ΑΠΕΙΡΟΣΤΙΚΟΣ ΙΙ
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Transcript of ΑΠΕΙΡΟΣΤΙΚΟΣ ΙΙ
-
Riemann
.
JJ II
J I
1 pi 273
pi url: http://www.aegean.gr
pi
pipi
832 00
Copyright pi ,
All rights reserved
-
Riemann
.
JJ II
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2 pi 273
1
1.1.
1.1.1 f : I R . F : I R
x I, F (x) = f (x), F pi f
f (x)dx. F (x) =
f (x)dx.
. F (x) pi f (x) H(x) = F (x)+c,c . H(x) = F (x) + c f .
1)
xndx = x
n+1
n+1 + c, n N2)
dxx = ln(x) + c
3)xadx = x
a+1
a+1 + c, a R
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Riemann
.
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3 pi 273
4)
sin(x)dx = cos(x) + c5)
cos(x)dx = sin(x) + c
6)
dxcos2(x) = tan(x) + c
7)
dxsin2(x) = cot(x) + c
8)axdx = a
x
ln(a) + c
9)exdx = ex + c
10)
dx1x2 = arcsin(x) + c
11)
dx1+x2
= arctan(x) + c
12)
sinh(x)dx = cosh(x) + c12)
cosh(x)dx = sinh(x) + c
1.1.2 f 1, f 2 : I R pi - pi h(x) = c1f1(x) + c2f2(x) pi c1, c2 R
(c1f1(x) + c2f2(x)) = c1
f1(x)dx + c2
f2(x)dx
1.1.3 1, 2 : 1 2 , f : 2 R - (t) , 0.
f (x)dx =
f ((t))(t)dt
: pi , pi pi, pi pi.
pi pia2 x2 x = |a| sin(u)
x = |a| cos(u).
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pi pia2 + x2 x = |a| tan(u)
x = |a| cot(u). pi pi
x2 a2 x = |a| 1cos(u) x =|a| cosh(u).
pi piax + b t = ax + b.
pi pi2ax x2 x = a(1 cos(u)).
1.1.4 f, g : R , pi f , g pi
f (x)g(x)dx = f (x)g(x)
f (x)g(x)dx, x
: I =
P(x)eaxdx, a R, P(x) pi
I =
P(x)eaxdx =
1a
P(x)deax
=1aP(x)eax 1
a
P (x)eaxdx.
I =P(x) cos(ax + b)dx, a, b R, P(x) pi
I =
P(x) cos(ax + b)dx =
1a
P(x)d sin(ax + b)
=1aP(x) sin(ax + b) 1
a
P (x) sin(ax + b)dx.
( cos sin.)
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.
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5 pi 273
I =P(x)ekx cos(ax + b)dx, a, b R, P(x) pi-
f (x) =ekx sin(ax + b)dx I =
P(x)df (x) = P(x)f (x) f (x)P (x)dx.
( cos sin.)
I =
ekx cos(ax + b) sin(ax + b)dx I =
ekx sin(ax + b) sin(ax + b)dx
I =ekx cos(ax + b) cos(ax + b)dx, a, b R pi
2 sin(a) sin(b) = cos(a b) cos(a + b),2 sin(a) cos(b) = sin(a + b) + sin(a b),2 cos(a) cos(b) = cos(a + b) + cos(a b).
I =f (x) ln((x))dx I =
f (x) arcsin((x))dx
I =f (x) arccos((x))dx
f (x) arctan((x))dx, pi f ,
F =f
f (x) ln((x))dx =
ln((x))df (x) = F (x) ln((x)) F (x)(x)(x) dx.
I =
f (x)2(x)dx, pi
1(x)
.
1.1.5 f : R R f (x) = p(x)q(x) pip, q pi.
: R , R(x) = P(x)Q(x) pi m 0 P(x) n
Q(x) 0. m n P(x) = P1(x)Q(x) + P2(x) pi piP1(x) P2(x). P(x)Q(x) = P1(x) +
P2(x)Q(x) , pi deg(P2(x)) n. pi
pi pipi :
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.
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m n pipi pi Q(x) pi R(x) pi pi pi Q(x). Q(x) pi pi P(x)Q(x) =
A1(xr1) +
A2(xr1) + . . .
An(xrn ) .
Q(x) pipi pi
P(x)Q(x)dx =
(x)Q1(x)
+1(x)Q2(x)
dx,pi Q1(x) = (Q(x), Q(x)), Q2(x) = Q(x)Q1(x) deg((x)) deg(Q1(x)) 1,deg(1(x)) deg(Q2(x)) 1. ( (x) 1(x) pi pi-
P(x)Q(x)dx =
(x)Q1(x)
+
1(x)Q2(x)
dx.
pi pi - , .
:
R
(x, n
ax+bcx+d
)dx, ( ax+bcx+d > 0 n )
ax+bcx+d =
tn.
R
(x,ax2 + bx + c
)dx :
a > 0 ax2 + bx + c = t ax ax2 + bx + c = t + ax.
a < 0 = b2 4ac > 0 ax2 + bx + c = t |x r1| pi r1 ax2 + bx + c = 0. a < 0 c > 0
ax2 + bx + c = tx c ax2 + bx + c = tx + c.
1.2.
1.2.1 pi ((x) + 1)(x +
(x) + 1)dx.
pi-
1.2.2 pi x2ex
3dx. pi-
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1.2.3 pi
1ex+1dx. pi-
1.2.4 pi
x+22x1dx. pi-
1.2.5 pi
1sin(x)dx. pi-
1.2.6 pi
1 x2dx. pi-
1.2.7 pi
1(xa)(bx)dx a < b.
pi-
1.2.8 pi
(x a)(b x)dx a < b .pi-
1.2.9 pi
ex
e2x+1dx. pi-
1.2.10 pi
13+x2dx. pi-
1.2.11 pi
x2 5dx. pi-
1.2.12 pi
xxx2dx. pi-
1.2.13 pi
cos(ax + b)dx. pi-
1.2.14 pi
5x+3x2+2x3dx. pi-
1.2.15 pi
11+cos(x)dx. pi-
1.2.16 pi
x4+x2
dx. pi-
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1.2.17 pi
1xcos(x)dx. pi-
1.2.18 pi
2x2+5x1x3+x22x dx. pi-
1.2.19 pi
x2+2x+3(x1)(x+1)2dx. pi-
1.2.20 pi
3x2+2x2x31 dx. pi-
1.2.21 pi
1(x31)2dx. pi-
1.2.22 pi x2exdx. pi-
1.2.23 a b pi pi - C(x) =
eax cos(bx)dx S(x) =
eax sin(bx)dx. pi-
1.2.24 pi x2 cos(x)dx. pi-
1.2.25 pi
1x ln(x)dx, x > 1. pi-
1.2.26 pi
ln(x)x dx, x > 1. pi-
1.2.27 pi
ln(x)dx. pi-
1.2.28 pi (x2 1) cos(3x)dx. pi-
1.2.29 pi xex cos(x)dx. pi-
1.2.30 pi ex sin(2x) cos(x)dx. pi-
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1.2.31 pi
3x212x
xarctan(x)dx. pi-
1.2.32 In =
1(ax2+bx+c)n dx
In+1 =2ax+b
n(4acb2)(ax2+bx+c)n +2(2n1)an(4acb2) In. pi-
1.2.33 pi (
1+x1x
) 12 dx. pi-
1.2.34 pi
143xx2dx. pi-
1.2.35 pi
1x+
x2x1dx. pi-
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2
Riemann
2.1.
2.1.1 I = [a. b] f : I R . P = {x0, x1, x2, . . . , xn} a = x0 < x1 < x2 < . . . 0, pi > 0 |P | < = |L(P, f ) U (P, f )| < 2.1.8 f : [a, b] R Riemann, baf (x)dx, pi.
Riemann
2.1.9 f : [a, b] R c R bacf (x)dx =
c baf (x)dx.
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2.1.10 f, g : [a, b] R bag(x) + f (x)dx =
baf (x)dx +
bag(x)dx.
2.1.11 f : [a, b] R c R baf (x)dx = c
af (x)dx +
bcf (x)dx.
2.1.12 f : [a, b] R | baf (x)dx | b
a|f (x)|dx.
2.1.13 f : [a, b] R F (x) = xaf (t)dt
2.1.14 f : [a, b] R pi (a, b) baf (x)dx = f ( )(b a).
2.1.15 f : [a, b] R F (x) = xaf (t)dt
x [a, b] F (x) = f (x). 2.1.16 f : [a, b] R H : [a, b] R pi f(H = f, x [a, b]), b
af (t)dt = H(a) H(b).
pi
2.1.17 f, g : [a, b] R baf (x)g(x)dx =
f (x)g(x)|ba baf (x)g(x)dx
2.1.18 g : [a, b] R pi f :[g(a), g(b)] R g(b)
g(a) f (x)dx = baf (g(x))g(x)dx.
.
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.
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2.1.19 f, g : [a, b] R . g pi pi [a, b] f ( ) b
ag(x)dx =
baf (x)g(x)dx.
2.1.20 f : [a, b] R G(x) = bxf (t)dt pi
G(x) = f (x) x [a, b]. 2.1.21 f, g : [a, b] R . g pi f pi [a, b] b
af (x)dx = f (a)
ag(x)dx +
f (b) bg(x)dx.
Bonnet 2.1.22 f, g : [a, b] R . f , g pi pi [a, b] b
af (x)g(x)dx =
f (a) ag(x)dx.
2.1.23 f, g : [a, b] R . f , g pi pi [a, b] b
af (x)g(x)dx =
f (b) bg(x)dx.
b
af (x)dx ' x ni=1 yi , yi = f (xi) = f (a + i ban ) pi b
af (x)dx ' x2 (y0 + 2y1 + 2y2 + . . . + 2yn1 + yn). Simpson b
af (x)dx ' x3 {(y0 + yn) + 4(y1 + y3 + . . . + yn1) + 2(y2 + y4 + . . . + yn)}.
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pipi f 0, f : [a, b] R A pi pi pi pi f xx x = a, x = b A =
baf (x)dx.
f A = ba|f (x)|dx.
f, g : [a, b] R 0 g(x) f (x) A(R) R = {f, g, x = a, x = b} A = b
af (x)dx b
ag(x)dx =
ba[f (x) g(x)]dx.
x = g(t), y = f (t), t [t1, t2] g(t) , 0 x = g(t), y = f (t) y x
2.1.24 g(x) : [a, b] R x = g(t), y = f (t) E =
bag(x)dx =
t2t1f (t)g(t)dt, g(t1) = a, g(t2) = b f, g [t1, t2]
pi
2.1.25 pi pi x = g(t), y = f (t), t [a, b] g, f [a, b] S = L() =
ba
g(t)2 + f (t)2dt.
pi pi pi
2.1.26 pi pi x = g(t), y = f (t), t [a, b] g, f [a, b] pi pi pi xx B = 2pi
ba|f (t)|g(t)2 + f (t2)dt.
pi y = f (x), x [a, b] B = 2pi ba|f (t)|1 + f (x)2dx
pi pi f : [a, b] R R = {f, Ox, x = a, x = b} pi pi
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f pi Ox x = a, x = b, V = pi
baf (x)2dx
f, g : [a, b] R 0 g(x) f (x) pi pi pipi f g, R = {f, g, Ox, x = a, x = b} V = pi
ba{f (x)2 g(x)2}dx.
x = g(t), y = f (t), t = [t1, t2] V = pi t2t1{f (t)2g(t)}dt g(t1) = a, g(t2) = b.
2.2.
2.2.1 pi Riemann
limn
1n
nk=1
nek
pi-
2.2.2 f, g, h : [a, b] R f (t) g(t) h(t), t [a, b]. f, h R([a, b]), pi R([a, b]) pi Riemann [a, b],
bafdx =
bahdx g Riemann -
, g R([a, b]) bagdx =
bafdx. pi-
2.2.3 pi f : [a, b] R Riemann pi
ba|f (t)|dt = 0 f = 0. pi-
2.2.4 pi n = 1,2, 12 +
13 + +
1n a > 0. pi-
2.2.7 f : [a, b] R pi
limn
b an
nk=1
f(a + k
b an
)=
baf (x)dx.
pi-
2.2.8 f : [a, b] R limn 1nn
i=1 f (in )
. pi-
2.2.9 pi
an =1
n + 1 +1
n + 2 + +1
n + n
pi-
2.2.10 pi 10 e
xdx. pi-
2.2.11 f [a, b] pi c [a, b] f (c) > 0,
baf (x)dx > 0. pi-
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2.2.12 pi )
10 x
2dx
) 10 x
3dx
) pi0 cos(x)dx
) 11(2x
2 x3)dx) 32 e
x/2dxpi-
2.2.13 pi pi
pi0 (x + pi) sin(x)dx.
pi-
2.2.14 ba
sin(x)x dx, 0 < a < b 0
pi-
2.2.15 pi pi - f (x) = x3 x2 x + 1, g(x) = x + 1. pi- 2.2.16 pi pi pi y =
1 sin(x) [0,2pi]. pi-
2.2.17 pi pi piy = x2, x + y = 2. pi-
2.2.18 pi pi r =6 sin() r = 2(1 cos()) pi- 2.2.19 pi pi y = x2, 0 x 1.pi-
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.
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2.2.20 pi pix = cos(t)(1 + cos(t)), y = sin(t)(1 + cos(t)), t [0,2pi]. pi-
2.2.21 pi 41 (2x
3 5x)dx pi Riemann pi-
2.2.22 pi 40
3x + 4dx. pi-
2.2.23 pi pi pi pi y =sin(x), y = cos(x), x = 0,x = pi/2 pi-
2.2.24 pi pi a, b, b pi pi pi pi x = a cos(t),y = b sin(t), t [0, pi], (a > b) pi x-. pi-
2.2.25 pi pi pi pi pi x = a(t sin(t)), y = a(1 cos(t)), t [0,2pi] pi x-.pi-
2.2.26 pi pi pi pi pi pi x = t2, y = t3 (t
2 3), pi x-. pi-
2.2.27 pi pi pi pi pi ,pi pi pi y = x2 y = x + 2, pi x - .pi-
2.2.28 pi pi pi pi pi ,pi pi x
2
a2 +y2
a2 = 1 pi x - . pi-
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.
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2.2.29 pi pi pi pi pi -, x = a cos3(t), y = a sin3(t) pi x - . pi-
2.2.30 pi pi 10
1 + x4dx -
pi pi-
2.2.31 pi pi 10
1 + x4dx -
Simpson pi-
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.
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3
3.1.
(a)N pi . G = i=1 ai (G)N .
3.1.1 (a)N R (G)N . (a, G)
=1 a.
3.1.2
=1 a l R,
=1 a = l lim G = l l = + pi . l = pi . pi pi.
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.
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3.1.3
=1 a (G)N (a)N .
3.1.4
=1 a = l
=1 b = m, l, m R
=1 a +nb = l +nm.
=1 a pi pi k pi
=1 a a > 0, N .
3.1.5 (G)N .
3.1.6
=1 a f : [1, +) R f () = a, N =1 a 1 f (x)dx .
.
3.1.7
=1 a,
=1 b a b, N1)
=1 a +
=1 a
2)
=1 a +
=1 b .
3.1.8
=1 a, l = lima+1a
1) l < 1 =1 a 2) l > 1 =1 a pi .
. pi .
-
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.
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3.1.9
=1 a .
3.1.10
=1 a, (a)N lim a = 0.
3.1.11
=1 a pi
=1 |a | . 3.1.12 pi .
3.1.13 ( - )
=1 a 0 N : a q 0, pi a q > 1, > 0.
3.2.
3.2.1 n=1 n2n (n+3) .pi-
3.2.2 pi n=1 3nn!pi-
3.2.3 n=1 1n pi.pi-
3.2.4 pi p Z x R n=1 npxnn! .pi-
3.2.5 n=1 n+2n3 , pi .pi-
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.
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3.2.6 pi n=1 3nn10 .pi-
3.2.7 pi n=1 n5n/2 .pi-
3.2.8 pi n=1 9nn! .pi-
3.2.9 pi n=1 nn(3n+1)n .pi-
3.2.10 pi n=1 1+cos2(nx)2n .pi-
3.2.11 pi limn (n!)n
nn2= 0.
pi-
3.2.12 pi n=1 11+2n1 .pi-
3.2.13 pi n=1 2n1n .pi-
3.2.14 pi n=1 12n1 .pi-
3.2.15 n=1 1na a > 1.pi-
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3.2.16 pi n=1 n2n .pi-
3.2.17 pi n=1 nnn! .pi-
3.2.18 pi n=1 (1 + 1n )n2 .pi-
3.2.19 pi n=1 ( 12n + 13n ) - pi .
pi-
3.2.20 limn n1357(2n1) = 0pi-
3.2.21 (an)nN 0 an 9, n N. n=1 an10n a 0 a 1.
pi-
3.2.22 pi n=1 5n7nn 32 .pi-
3.2.23 pi n=1 sin( 1n ).pi-
3.2.24 n=1 1n(ln(n))a a > 1.pi-
3.2.25 pi pi pi n=1(1)n n2+12n3+n1 .pi-
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.
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3.2.26 pi pi pi n=1 (1)n+12n .pi-
3.2.27 pi n=1 2+(1)n2n .pi-
3.2.28
1 ln(21 ) +12 ln(
32 ) + +
1n ln(n + 1
n)
.pi-
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Riemann
.
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4
4.1.
4.1.1 f : [a,+) R . f pi a +, a
f (x)dx = limx xaf (t)dt.
limx xaf (t)dt pi f .
limx xaf (t)dt + () f pi
(). pi .
b f (x)dx = limx
bxf (t)dt
f (x)dx = limx
xcf (t)dt +
limx cxf (t)dt
-
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.
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4.1.2 a
f (t)dt+ a
g(t)dt pi a
lf (t)+mg(t)dt = l a
f (t)dt+m
a
g(t)dt.
. 4.1.3 f : [a,+) R
af (x)dx pi
> 0, pi > 0 y1, y2 [a,+), x1, x2 > ,| x2
x1f (x)dx | 0, pi > 0 q1, q2 [a, b], |x1 b| < , |x2 b| < | x2
x1f (x)dx | < .
4.1.12 f : [a,+b) R+ F (x) = xa|f (t)|dt
baf (x)dx .
4.1.13 f, g : [a,+b) R 0 f (x) g(x),a x < b 1)
bag(x)dx < b
af (x)dx < .
2) bag(x)dx = b
ag(x)dx =
-
Riemann
.
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4.1.14 f, g : [a,+) R f (x) 0,g(x) 0 x [a,+b) limx f (x)g(x) = l 1) 0 < l < b
af (x)dx < , b
ag(x)dx <
2) l = 0 bag(x)dx < , b
af (x)dx <
3) l = baf (x)dx = , b
ag(x)dx = .
4.1.15
n=0 anxn an pi x.
n=0 an(x a)n an pi x a
4.1.16 I x I .
.
4.1.17 Sn(x) =n1
m=0 amxm -
Rn(x) =
m=n amxm pipi .
n=0
anxn = Sn(x) + Rn(x).
4.1.18 x = x0
n=0 anxn
S(x0) = limn Sn(x).
n=0 anx
n x = x0 > 0, > 0 : n n0 pin0 Rn(x0) < 0.
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Riemann
.
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4.2.
4.2.1 I = a
dxxk , a > 0 k > 1
pi k 1.pi-
4.2.2 pi 0
11+x2dx.
pi-
4.2.3 pi
11+x2dx.
pi-
4.2.4 pi )
1
x1+x2
) 0 cos(x)dx
) 0 e
xdx)
1
ln(x)x
pi-
4.2.5 0 e
x2dx .pi-
4.2.6 1
sin(x)xa dx
1
cos(x)xa dx pi-
a > 1 a > 0.pi-
4.2.7 pi 3
dxx2+x2 .
pi-
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Riemann
.
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4.2.8 pi 1
x
(1+x2)dx =12 +
pi2 .
pi-
4.2.9 a
sin(x)x dx pi -
Cauchypi-
4.2.10 a
11+x4dx, a > 0 .
pi-
4.2.11 0
1(1+x3)1/3dx, .
pi-
4.2.12 0
x2
2x4x2+1dx, .pi-
4.2.13 0 x sin(x
4)dx, .pi-
4.2.14 ba
dx(xa)k k < 1 pi-
k > 1.pi-
4.2.15 ba
dx(xa)(bx) .
pi-
4.2.16 10 sin(
1x )dx .
pi-
4.2.17 pi p pi0
dxsinp(x) .
pi-
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Riemann
.
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4.2.18 pi
(x) = +0
tx1etdt,
x > 0. ( )pi-
4.2.19 pi
B(x, y) = 10tx1(1 t)y1dt,
x > 0, y < 1. ( )pi-
4.2.20 pi 0
cos(x)x
dx,
.pi-
4.2.21 pi 10
11x2dx .
pi-
4.2.22 n=1 n+1n2+1 (x2)n.
pi-
-
Riemann
.
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J I
33 pi 273
4.2.23 n=1 3n2n+4 xn.pi-
4.2.24 n=1 xnn! .pi-
4.2.25 n=1 nn(x1)n.
pi-
4.2.26 n=1(1)n+1 xnn .pi-
4.2.27 pi n=1 xn ln(x)n .pi-
4.2.28 limn nan = a,
n=1 anxn R = 1a .
pi-
4.2.29 an .
n=0 an
2n=0
an +n=1
nanxn = 0,
an.pi-
4.2.30 pi n=0 anxn R pi n=0 anx2n
pi-
-
Riemann
.
JJ II
J I
34 pi 273
4.2.31 pi n=1 naxn.pi-
-
Riemann
.
JJ II
J I
35 pi 273
pi : pi pi .
1.2.1
-
Riemann
.
JJ II
J I
36 pi 273
pi : pi -
(x)e(x)dx
1.2.2
-
Riemann
.
JJ II
J I
37 pi 273
pi : 1ex+1 = 1 ex
1+ex .
1.2.3
-
Riemann
.
JJ II
J I
38 pi 273
pi : a b x+22x1 = a +b
2x1 .
1.2.4
-
Riemann
.
JJ II
J I
39 pi 273
pi : pi sin(x) = 2 sin( x2 ) cos(x2 ).
1.2.5
-
Riemann
.
JJ II
J I
40 pi 273
pi : x = sin(y).
1.2.6
-
Riemann
.
JJ II
J I
41 pi 273
pi : x = a + (b a) sin2(u).
1.2.7
-
Riemann
.
JJ II
J I
42 pi 273
pi : x = a + (b a) sin2(u).
1.2.8
-
Riemann
.
JJ II
J I
43 pi 273
pi : ex = y.
1.2.9
-
Riemann
.
JJ II
J I
44 pi 273
pi : x =3 tan(u).
1.2.10
-
Riemann
.
JJ II
J I
45 pi 273
pi : x =5 cosh(u).
1.2.11
-
Riemann
.
JJ II
J I
46 pi 273
pi : x = 12 (1 cos(u)).
1.2.12
-
Riemann
.
JJ II
J I
47 pi 273
pi : t = ax + b.
1.2.13
-
Riemann
.
JJ II
J I
48 pi 273
pi : a b 5x+3x2+2x3 =a
x1 +b
x+3 pi pi pipi.
1.2.14
-
Riemann
.
JJ II
J I
49 pi 273
pi : x = 2 arctan(t).
1.2.15
-
Riemann
.
JJ II
J I
50 pi 273
pi : pi x = 2 tan(u).
1.2.16
-
Riemann
.
JJ II
J I
51 pi 273
pi : pi x = t.
1.2.17
-
Riemann
.
JJ II
J I
52 pi 273
pi : pi pi pi a, b, c 2x2+5x1x3+x22x =
ax +
bx1 +
cx+2
1.2.18
-
Riemann
.
JJ II
J I
53 pi 273
pi : pi pi pi a, b, c x2+2x+3
(x1)(x+1)2 =a
x1 +b
x+1 +c
(x+1)2 .
1.2.19
-
Riemann
.
JJ II
J I
54 pi 273
pi : pi pi pi a, b, c 3x2+2x2
x31 =3x2+2x2
(x1)(x2+x+1) =a
x1 +bx+c
x2+x+1 .
1.2.20
-
Riemann
.
JJ II
J I
55 pi 273
pi : pi
P(x)Q(x)dx =
(x)Q1(x)
+
1(x)Q2(x)
dx (x),1(x), Q1(x), Q2(x).
1.2.21
-
Riemann
.
JJ II
J I
56 pi 273
pi : pi .
1.2.22
-
Riemann
.
JJ II
J I
57 pi 273
pi : pi - S(x), C(x).
1.2.23
-
Riemann
.
JJ II
J I
58 pi 273
pi : pi pi .
1.2.24
-
Riemann
.
JJ II
J I
59 pi 273
pi : pi pi .
1.2.25
-
Riemann
.
JJ II
J I
60 pi 273
pi : pi pi .
1.2.26
-
Riemann
.
JJ II
J I
61 pi 273
pi : pi pi 1 = (x).
1.2.27
-
Riemann
.
JJ II
J I
62 pi 273
pi : pi pi .
1.2.28
-
Riemann
.
JJ II
J I
63 pi 273
pi : f (x) =ex cos(x)dx pi pi .
1.2.29
-
Riemann
.
JJ II
J I
64 pi 273
pi : pi sin(2x) cos(x) = 12 (sin(3x) + sin(x)), pi pi pi pipi.
1.2.30
-
Riemann
.
JJ II
J I
65 pi 273
pi : pi
3x212x
xdx pi pi-
.
1.2.31
-
Riemann
.
JJ II
J I
66 pi 273
pi : pi .
1.2.32
-
Riemann
.
JJ II
J I
67 pi 273
pi : pi (1+x) pi :
(1+x1x
) 12=
1+x(1x)(1x) =
1(1x2) +
x(1x2) pi pi pipi -
.
1.2.33
-
Riemann
.
JJ II
J I
68 pi 273
pi : 4 3x x2 = t(x + 4).
1.2.34
-
Riemann
.
JJ II
J I
69 pi 273
pi : x2 x 1 = t x.
1.2.35
-
Riemann
.
JJ II
J I
70 pi 273
pi : pi pi pi . pi Pn pi.. U (f, Pn) .
2.2.1
-
Riemann
.
JJ II
J I
71 pi 273
pi : Pn limn[U (f, Pn) L(g, Pn)] = 0.
2.2.2
-
Riemann
.
JJ II
J I
72 pi 273
pi : f , 0 ba|f |dt pi .
2.2.3
-
Riemann
.
JJ II
J I
73 pi 273
pi : Pn pi.. U ( 1x , Pn) = 1 +12 + 1n1 .
2.2.4
-
Riemann
.
JJ II
J I
74 pi 273
pi : Riemann pi .
2.2.5
-
Riemann
.
JJ II
J I
75 pi 273
pi : Riemann pi .
2.2.6
-
Riemann
.
JJ II
J I
76 pi 273
pi : Riemann.
2.2.7
-
Riemann
.
JJ II
J I
77 pi 273
pi : pi limn bann
k=1 f(a + k ban
)=
baf (x)dx. -
a b.
2.2.8
-
Riemann
.
JJ II
J I
78 pi 273
pi : pi limn bann
k=1 f(a + k ban
)=
baf (x)dx.
2.2.9
-
Riemann
.
JJ II
J I
79 pi 273
pi : Riemann.
2.2.10
-
Riemann
.
JJ II
J I
80 pi 273
pi : pi f f (x) > 0 pi c.
2.2.11
-
Riemann
.
JJ II
J I
81 pi 273
pi : pi pi .
2.2.12
-
Riemann
.
JJ II
J I
82 pi 273
pi : f (x) = x +pi, g(x) = sin(x) pipi f g.
2.2.13
-
Riemann
.
JJ II
J I
83 pi 273
pi :
2.2.14
-
Riemann
.
JJ II
J I
84 pi 273
pi : f (x) g(x) pi pi f (x) g(x) .
2.2.15
-
Riemann
.
JJ II
J I
85 pi 273
pi : pi 2pi0
1 sin(x)dx.
2.2.16
-
Riemann
.
JJ II
J I
86 pi 273
pi : pi y = x2, x + y = 2
2.2.17
-
Riemann
.
JJ II
J I
87 pi 273
pi : pi.
2.2.18
-
Riemann
.
JJ II
J I
88 pi 273
pi : pi pi pi -
ba
x (t)2 + y(t)2dt.
2.2.19
-
Riemann
.
JJ II
J I
89 pi 273
pi : pi ba
x (t)2 + y(t)2dt.
2.2.20
-
Riemann
.
JJ II
J I
90 pi 273
pi : pi limn bann
k=1 f(a + k ban
)=
baf (x)dx.
2.2.21
-
Riemann
.
JJ II
J I
91 pi 273
pi : u = 3x + 4.
2.2.22
-
Riemann
.
JJ II
J I
92 pi 273
pi : pi pi pi pi pi x = 0 x = pi/2.
2.2.23
-
Riemann
.
JJ II
J I
93 pi 273
pi : pi pi pi pi x - pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),
l t m.
2.2.24
-
Riemann
.
JJ II
J I
94 pi 273
pi : pi pi pi pi x - pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),
l t m.
2.2.25
-
Riemann
.
JJ II
J I
95 pi 273
pi : pi pi pi pi x - pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),
l t m.
2.2.26
-
Riemann
.
JJ II
J I
96 pi 273
pi : pi pi pi f , g, f (x) g(x) 0, x [a, b] V = pi b
a{f (x2) g(x)2}dx.
2.2.27
-
Riemann
.
JJ II
J I
97 pi 273
pi : pi pi pi f , x [a, b] V = pi b
af (x)2dx.
2.2.28
-
Riemann
.
JJ II
J I
98 pi 273
pi : pi pi pi - x = g(t), y = f (t), t [t1, t2] V = pi
t2t1f (t)2g(t)dt.
2.2.29
-
Riemann
.
JJ II
J I
99 pi 273
pi : [0,1] 10 pi - pi.
2.2.30
-
Riemann
.
JJ II
J I
100 pi 273
pi : [0,1] 10 pi - Simpson.
2.2.31
-
Riemann
.
JJ II
J I
101 pi 273
pi : pi .
3.2.1
-
Riemann
.
JJ II
J I
102 pi 273
pi : pi .
3.2.2
-
Riemann
.
JJ II
J I
103 pi 273
pi : pi .
3.2.3
-
Riemann
.
JJ II
J I
104 pi 273
pi : pi .
3.2.4
-
Riemann
.
JJ II
J I
105 pi 273
pi : pi n=1 1nk .
3.2.5
-
Riemann
.
JJ II
J I
106 pi 273
pi : pi .
3.2.6
-
Riemann
.
JJ II
J I
107 pi 273
pi : pi .
3.2.7
-
Riemann
.
JJ II
J I
108 pi 273
pi : pi .
3.2.8
-
Riemann
.
JJ II
J I
109 pi 273
pi : pi .
3.2.9
-
Riemann
.
JJ II
J I
110 pi 273
pi : pi .
3.2.10
-
Riemann
.
JJ II
J I
111 pi 273
pi : n=1 (n!)nnn2 .
3.2.11
-
Riemann
.
JJ II
J I
112 pi 273
pi : pi .
3.2.12
-
Riemann
.
JJ II
J I
113 pi 273
pi : pi .
3.2.13
-
Riemann
.
JJ II
J I
114 pi 273
pi : pi .
3.2.14
-
Riemann
.
JJ II
J I
115 pi 273
pi : pi .
3.2.15
-
Riemann
.
JJ II
J I
116 pi 273
pi : pi .
3.2.16
-
Riemann
.
JJ II
J I
117 pi 273
pi : pi .
3.2.17
-
Riemann
.
JJ II
J I
118 pi 273
pi : pi .
3.2.18
-
Riemann
.
JJ II
J I
119 pi 273
pi : pi .
3.2.19
-
Riemann
.
JJ II
J I
120 pi 273
pi : n=1 n1357(2n1) .
3.2.20
-
Riemann
.
JJ II
J I
121 pi 273
pi : pi .
3.2.21
-
Riemann
.
JJ II
J I
122 pi 273
pi : pi .
3.2.22
-
Riemann
.
JJ II
J I
123 pi 273
pi : pi .
3.2.23
-
Riemann
.
JJ II
J I
124 pi 273
pi : pi .
3.2.24
-
Riemann
.
JJ II
J I
125 pi 273
pi : pi .
3.2.25
-
Riemann
.
JJ II
J I
126 pi 273
pi : pi .
3.2.26
-
Riemann
.
JJ II
J I
127 pi 273
pi : Sn =n
k=122k +
nk=1
(1)k2k .
3.2.27
-
Riemann
.
JJ II
J I
128 pi 273
pi : pi .
3.2.28
-
Riemann
.
JJ II
J I
129 pi 273
pi : pi .
4.2.1
-
Riemann
.
JJ II
J I
130 pi 273
pi : pi .
4.2.2
-
Riemann
.
JJ II
J I
131 pi 273
pi : pi .
4.2.3
-
Riemann
.
JJ II
J I
132 pi 273
pi : pi .
4.2.4
-
Riemann
.
JJ II
J I
133 pi 273
pi : pi .
4.2.5
-
Riemann
.
JJ II
J I
134 pi 273
pi : | sin(x)|xa 1xa | cos(x)|xa 1xa x 1.
4.2.6
-
Riemann
.
JJ II
J I
135 pi 273
pi : pi t3
dxx2+x2 pi pi t
4.2.7
-
Riemann
.
JJ II
J I
136 pi 273
pi : pi .
4.2.8
-
Riemann
.
JJ II
J I
137 pi 273
pi : > 0 x1, x2 > | x2x1
sin(x)x dx | <
4.2.9
-
Riemann
.
JJ II
J I
138 pi 273
pi : pi .
4.2.10
-
Riemann
.
JJ II
J I
139 pi 273
pi : pi .
4.2.11
-
Riemann
.
JJ II
J I
140 pi 273
pi : pi .
4.2.12
-
Riemann
.
JJ II
J I
141 pi 273
pi : pi Cauchy.
4.2.13
-
Riemann
.
JJ II
J I
142 pi 273
pi : pi .
4.2.14
-
Riemann
.
JJ II
J I
143 pi 273
pi : k < 1 limxa(x a)kf (x) = A < +
baf (x)dx .
4.2.15
-
Riemann
.
JJ II
J I
144 pi 273
pi : pi x = 1t pi pi pipi.
4.2.16
-
Riemann
.
JJ II
J I
145 pi 273
pi : pi .
4.2.17
-
Riemann
.
JJ II
J I
146 pi 273
pi : pi .
4.2.18
-
Riemann
.
JJ II
J I
147 pi 273
pi : pi .
4.2.19
-
Riemann
.
JJ II
J I
148 pi 273
pi : pi .
4.2.20
-
Riemann
.
JJ II
J I
149 pi 273
pi : pi .
4.2.21
-
Riemann
.
JJ II
J I
150 pi 273
pi : .
4.2.22
-
Riemann
.
JJ II
J I
151 pi 273
pi : .
4.2.23
-
Riemann
.
JJ II
J I
152 pi 273
pi : .
4.2.24
-
Riemann
.
JJ II
J I
153 pi 273
pi : .
4.2.25
-
Riemann
.
JJ II
J I
154 pi 273
pi : .
4.2.26
-
Riemann
.
JJ II
J I
155 pi 273
pi : .
4.2.27
-
Riemann
.
JJ II
J I
156 pi 273
pi : .
4.2.28
-
Riemann
.
JJ II
J I
157 pi 273
pi : f (x) = n=0 an pi f (x) pi 2f (x) + f (x) = 0 .
4.2.29
-
Riemann
.
JJ II
J I
158 pi 273
pi : pi |x2| < R.
4.2.30
-
Riemann
.
JJ II
J I
159 pi 273
pi : pi
4.2.31
-
Riemann
.
JJ II
J I
160 pi 273
pi : (x + 1)(x +
x + 1) = 2x + 2
x + x
32 + 1.
(x + 1)(x x + 1)dx =
(2x + 2
x + x
32 + 1)dx
=
2xdx +
2xdx +
x
32dx +
dx
= x2 +1x+25x
52 + x + C.
1.2.1
-
Riemann
.
JJ II
J I
161 pi 273
pi : x2ex
3dx =
13
(x)e(x)dx,
pi y = (x) = x3, x2ex
3dx =
eydy =
ey
3 + C =ex
3
3 + C
1.2.2
-
Riemann
.
JJ II
J I
162 pi 273
pi : 1
1 + ex = 1 ex
1 + ex = 1 (1 + ex )
1 + ex ,
1ex + 1dx =
dx
(1 + ex )
1 + ex dx = x ln(1 + ex ) + C
1.2.3
-
Riemann
.
JJ II
J I
163 pi 273
pi :
x + 22x 1 =
122x + 42x 1 =
122x 1 + 52x 1 =
12 +
52
12x 1 .
x + 22x 1dx =
(12 +
52
12x 1 )dx
=12
dx +
52
12x 1dx
=x
2 +54 ln |2x 1| + C.
1.2.4
-
Riemann
.
JJ II
J I
164 pi 273
pi :
1sin(x)
dx =
12 sin( x2 ) cos(
x2 )dx
=
cos( x2 )sin( x2 ) cos
2( x2 )d(
x
2 ) = 1
tan( x2 ) cos2( x2 )
d(x
2 )
=
(tan( x2 ))tan( x2 )
d(x
2 ) = ln(tan(x
2 )) + C,
kpi < x < kpi + pi, k Z .
1.2.5pi : x = g(y) = sin(y), x [1, 1] pi y = arcsin(x), (y [ pi2 , pi2 ]) cos(y) 0 .
1 x2dx =
1 g(y)2g(y)dy =
cos(y)(sin(y))dx
=
cos2(y)dy =
1 + cos(2y)2 dy
=y
2 +sin(2y)
4 + c =y
2 +sin(y) cos(y)
2 + c
=arcsin(x)
2 +x1 x22 + c. (4.1)
-
Riemann
.
JJ II
J I
165 pi 273
1.2.6pi : pipi (x a)(b x) < 0 pi a < x < b 0 < xaba < 1.
xaba = sin
2(u)
x = g(u) = a + (b a) sin2(u), u (0, pi2 ),
b x = (b a) (x a) = (b a) (b a) sin2(u) = (b a) cos2(u), (x a)(b x) = (b a) sin(u) cos(u).
pi 1(x a)(b x)dx =
1(g(u) a)(b g(u))du =
=
1(b a) sin(u) cos(u) (b a)2 sin(u) cos(u))du
= 2
du = 2u + c = 2 arcsin(x ab x
) 12+ c.
1.2.7pi : x = a + (b a) sin2(u)
(x a)(b x)dx =
(b a) sin(u) cos(u)(b a)2 sin(u) cos(u)du
= 2(b a)2
sin2(u) cos2(u)du =(b a)2
2
sin2(2u)du.
-
Riemann
.
JJ II
J I
166 pi 273
sin2(2u)du = 12
sin2(v)dv (2u = v)
=14
(1 cos(2v))dv = 14v
18 sin(2v) + c
=12u
18 sin(4u) + c =
12u
14 sin(2u) cos(2u) + c.
=12u
12 sin(u) cos(u)(1 2 sin
2(u)) + c.
pi (x a)(b x) = (b a) sin(u) cos(u)
I =
(x a)(b x)dx
=(b a)2
2
sin2(2u)du
=(b a)2
4 u (b a)2
4 sin(u) cos(u)(1 2 sin2(u)) + c =
=(b a)2
4 arcsinx a
b a (b a)24
(x a)(b x)
b a (a 2x ab a ) + c
=(b a)2
4 arcsinx a
b a 14 (x a)(b x)(b a 2(x a)) + c
=(b a)2
4 arcsinx a
b a + 12 (x a)(b x)
(x b + a2
)+ c.
-
Riemann
.
JJ II
J I
167 pi 273
1.2.8pi : ex = y. exdx = dy
ex
e2x + 1dx =
y
y2 + 11ydy
=
1y2 + 1dy.
y = tan(u) dy = 1cos2(u)du. 1y2 + 1dx =
1tan2(u) + 1
1cos2(u)
du
=
du = u + C = arctan(ex ) + C.
1.2.9pi : x =
3 tan(u). dx =
3 1cos2(u)du 1
3 + x2dx = 1
31
tan2(u) + 11
cos2(u)du
=13
du =
13u + C =
13
arctan(x3) + C.
-
Riemann
.
JJ II
J I
168 pi 273
1.2.10pi : x =
5 cosh(u).
x2 5dx =
5 sinh(u)5 sinh(u)du
=
5 sinh2(u)du = 5
cosh(2z) 1
2 du =54 sinh(2u)
52u
=125 sinh(u)
5 sinh(u) 52u
=xx2 52
52 ln
x + x2 55
+ C.
1.2.11pi : x = 12 (1 cos(u)).
xx x2
dx =
x
2 12x x2dx
=
12 (1 cos(u))
12 sin(u)
12 sin(u)du =
12
(1 cos(u))du = 12 (u sin(u))
=12 arccos(1 2x)
x x2 + C.
-
Riemann
.
JJ II
J I
169 pi 273
1.2.12pi : t = ax + b.
cos(ax + b)dx =1a
cos(t)dt
=1a
sin(t) = 1a
cos(at + b) + C..
1.2.13pi : a b
5x + 3x2 + 2x 3 =
a
x 1 +b
x + 3 ,
x , 1 x , 3. 5x +3 = a(x +3)+b(x 1). x, x , 1 x , 3 pipi a = 2, b = 3.pi
5x + 3x2 + 2x 3dx = 2
1x 1dx + 3
1x + 3dx
= 2 ln |x 1| + 3 ln |x + 3| + c = ln |(x 1)2(x + 3)3| + c.
-
Riemann
.
JJ II
J I
170 pi 273
1.2.14pi :
x2 = arctan(t) dx =2
1+t2dt. 11 + cos(x)dx =
11 + 1t21+t2
21 + t2dt
=
11+t2+1t2
1+t2
21 + t2dt =
1 + t22
21 + t2dt
=
dt = t + C = tan(
x
2 ) + C.
1.2.15pi : x = 2 tan(u)
x
4 + x2dx =
2 tan(u)2
cos(u)
2cos2(u)
du =
tan(u)cos(u)
du
=
sin(u)cos2(u)
du
=
(cos(u))
cos2(u)= 2 1
cos(u)=4 + x2 + C.
1.2.16
-
Riemann
.
JJ II
J I
171 pi 273
pi : x = t
1x
cos(x)dx =
1tcos(t)2tdt = 2
cos(t)dt
= 2 sin(t) + C = 2 sin(x) + C
1.2.17pi : a, b c
2x2 + 5x 1x3 + x2 2x =
a
x+
b
x 1 +c
x + 2 ,
x , 0, x , 1 x , 2. pipi a = 12 , b = 2,c = 12 .pi
2x2 + 5x 1x3 + x2 2x dx =
12
1xdx + 2
1x 1dx
12
1x + 2dx
=12 ln |x | + 2 ln |x 1|
12 ln |x + 2| + C = ln |(x 1)
2
x
x + 2 | + C.
1.2.18
-
Riemann
.
JJ II
J I
172 pi 273
pi : a, b c x2 + 2x + 3
(x 1)(x + 1)2 =a
x 1 +b
x + 1 +c
(x + 1)2 ,
x , 1,1, pipi a = 32 , b = 12 , c = 1.pi
x2 + 2x + 3
(x 1)(x + 1)2dx =32
1x 1dx +
12
1x + 1dx
1(x + 1)2dx
=32 ln |x 1| +
12 ln |x + 1| +
1x + 1 + C = ln |
(x 1)3x + 1 | + C.
1.2.19pi : a, b c
3x2 + 2x 2x3 1 =
3x2 + 2x 2(x 1)(x2 + x + 1) =
a
x 1 +bx + c
x2 + x + 1 .
pipi a = 1, b = 2, c = 3.pi
3x2 + 2x 2x3 1 dx =
1x 1dx +
2x + 3x2 + x + 1dx
= ln |x 1| + 2x + 1
x2 + x + 1dx + 2 1
x2 + x + 1dx
= ln |x 1| + ln(x2 + x + 1) + 2 1
x2 + x + 1dx
-
Riemann
.
JJ II
J I
173 pi 273
1x2+x+1dx pi x
2 + x + 1 = (x + 12 )2 + 34 =
34 (y
2 + 1)pi y = f (x) = 2
(3) (x +12 ) f
(x) = 23 . 1x2 + x + 1dx =
1(x + 12 )
2 + 34dx =
f (x)
34 (f (x)
2 + 1)dx
=23
1y2 + 1dx =
23
arctan(y) + c.
3x2 + 2x 2x3 1 dx = ln |x 1| + ln(x
2 + x + 1) + 43
arctan(2x + 1
3) + C
1.2.20pi : pi
P(x)Q(x)dx =
(x)Q1(x)
+
1(x)Q2(x)
dx. Q1(x)={(x3 1)2,6x2(x3 1)} =
x3 1. pi (x) = Ax2 + Bx + , Q2(x) = x3 1, 1(x) = x2 + Ex + Z . 1(x3 1)2dx =
Ax2 + Bx +
x3 1 +
x2 + Ex + Z
x3 1 dx.
1(x3 1)2 =
(2Ax + B)(x3 1) 3x2(Ax2 + Bx + )(x3 1)2 +
x2 + Ex + Z
x3 1 ,
(2Ax + B)(x3 1) 3x2(Ax2 + Bx + ) + (x2 + Ex + Z )(x3 1) = 1. A = = = E = 0, B = 13 , Z = 23 . pi
1(x3 1)2 =
x
3(x3 1) 23
1x3 1dx.
-
Riemann
.
JJ II
J I
174 pi 273
1x31dx
1x31 =
lx1 +
mx+nx2+x+1 , pi l =
13 , m = 13 , n = 23 .
1x3 1dx =
13
1x 1dx
13
x + 2
x2 + x + 1dx
=13 ln(x 1)
16 ln(x
2 + x + 1) 13
arctan(2x + 1
3) + C.
1(x3 1)2dx =
x
3(x3 1) +19 ln
(x2 + x + 1(x 12)
)+
233
arctan(2x + 1
3) + C
1.2.21pi :
x2ex3dx =
x2(ex )dx = x2ex
(x2)exdx = x2ex 2
xexdx
= x2ex 2
x(ex )dx = x2ex 2xex + 2
(x)exdx
= x2ex 2xex + 2
1exdx = x2ex 2xex + 2ex + C.
1.2.22
-
Riemann
.
JJ II
J I
175 pi 273
pi :
aC(x) =
aeax cos(bx)dx =
(eax ) cos(bx)dx
= eax cos(bx)
eax (cos(bx))dx = eax
eax (b sin(bx))dx= eax cos(bx) + bS(x) + C1
pi
aC(x) bS(x) = eax cos(bx) + C1 (4.2) bC(x) + aS(x) = eax sin(bx) + C2. (4.3)
(4.2) (4.3)
C(x) =a cos(bx) + b sin(bx)
a2 + b2eax + C3 S(x) =
a sin(bx) b cos(bx)a2 + b2
eax + C4.
1.2.23pi : pi pi .
x2 cos(x)dx =
x2(sin(x))dx
= x2 sin(x) 2
x sin(x)dx = x2 sin(x) 2
x(cos(x))dx
= x2 sin(x) + 2x cos(x) 2
(x) cos(x)dx
= x2 sin(x) + 2x cos(x) 2 sin(x) + C.
-
Riemann
.
JJ II
J I
176 pi 273
1.2.24pi : 1
x ln(x)dx =
(ln(x))
ln(x)dx
= ln (ln(x)) + C.
1.2.25pi :
I =
ln(x)x
dx =
1xln(x)dx
=
(ln(x))ln(x)dx = ln2(x)
ln(x)x
dx = ln2(x) I.
2I = ln2(x) + C I = ln(x)x dx = ln2(x)2 + c 1.2.26
pi : ln(x)dx =
(x)ln(x)dx
= x ln(x)
x(ln(x))dx = x ln(x)
dx = x ln(x) x + C.
-
Riemann
.
JJ II
J I
177 pi 273
1.2.27
pi :
(x2 1) cos(3x)dx = 13 (x
2 1) sin(3x)
sin(3x)2xdx
=13 (x
2 1) sin(3x) + 29
x(cos(3x))dx
=13 (x
2 1) sin(3x) + 29x cos(3x) 29
cos(3x)dx
=13 (x
2 1) sin(3x) + 29x cos(3x) 227 sin(3x) + C.
1.2.28
-
Riemann
.
JJ II
J I
178 pi 273
pi :
I =
xex cos(x)dx
=
x
(ex
2 (sin(x) cos(x)))dx
=xex
2 (sin(x) cos(x)) 12
ex sin(x)dx +
12
ex (sin(x))dx
=xex
2 (sin(x) cos(x)) 12
ex sin(x)dx +
12e
x (sin(x)) 12
ex sin(x)dx
=xex
2 (sin(x) cos(x)) +12e
x cos(x) + C.
1.2.29
pi : pi sin(2x) cos(x) = 12 (sin(3x) + sin(x)) :
ex sin(2x) cos(x)dx = 12
ex sin(3x)dx + 12
ex sin(x)dx.
-
Riemann
.
JJ II
J I
179 pi 273
I1 =
ex sin(3x)dx = 13
ex (cos(3x))dx
=13e
x cos(3x) + 13
(ex ) cos(3x)dx = 13e
x cos(3x) + 19
ex (sin(3x))dx
=13e
x cos(3x) + 19ex sin(3x) 19
(ex ) sin(3x)dx
=13e
x cos(3x) + 19ex sin(3x) I1.
I1 = 110ex sin(3x) 3cos(3x). I2 =
ex sin(x) = e
x
2 (sin(x) cos(x)).
ex sin(2x) cos(x)dx = 120e
x sin(3x) 3cos(3x) + ex
4 (sin(x) cos(x)) + C.
1.2.30
pi : pi
3x212x
xdx. :
3x2 12x
xdx =
(32x 12x
32)dx = x
32 + x
12 =
x2 + 1x
.
-
Riemann
.
JJ II
J I
180 pi 273
3x2 12x
x
arctan(x)dx = (
x2 + 1x
)arctan(x)dx
=x2 + 1
xarctan(x)
x2 + 1
xarctan(x)dx
=x2 + 1
xarctan(x)
x2 + 1
x
11 + x2dx
=x2 + 1
xarctan(x) 2(x) + C.
1.2.31pi : :
In+1 =
1(ax2 + bx + c)n+1
dx =4a
4ac b2
ax2 + bx + c a(x + b2a )(ax2 + bx + c)n+1
dx
=4a
4ac b2 In
(2ax + b)2
(ax2 + bx + c)n+1dx
=4a
4ac b2 In 1
n(4ac b2)
(2ax + b)(
1(ax2 + bx + c)n
)dx
pi
=4a
4ac b2 In +1
n(4ac b2)2ax + b
(ax2 + bx + c)n 2an(4ac b2)
1(ax2 + bx + c)n
dx
=4a
4ac b2 In +1
n(4ac b2) (2ax + b)1
(ax2 + bx + c)n 2an(4ac b2)n In
-
Riemann
.
JJ II
J I
181 pi 273
1.2.32pi : (1 + x
1 x) 12dx =
1 + x(1 x)(1 x)dx
=
1(1 x2)
dx +
x
(1 x2)dx.
pi
1(1x2)dx.
x(1 x2)
dx = 12
(1 x2)(1 x2)
dx =
= 12 1
ydy (y = 1 x2)
= y + c = 1 x2 + c.
pi
1(1x2)dx = arcsin(x). (1 + x
1 x) 12dx = arcsin(x)
1 x2 + c
-
Riemann
.
JJ II
J I
182 pi 273
1.2.33pi :
R
(x,ax2 + bx + c
)dx a < 0 c > 0
4 3x x2 = t(x + 4) : 1
4 3x x2dx =
10t(1+t2)25t1+t2
dt =
= 2 1
1 + t2dt = 2 arctan(t) + C
= 2 arctan1 xx + 4
+ C.
1.2.34pi :
R
(x,ax2 + bx + c
)dx a > 0
x2 x 1 = t x : 1
x +x2 x 1
dx =
2t2 2t + 2t(2t 1)2 dt =
=
(2t 32t 1 +
3(2t 1)2
)dt
= 2 ln(|t |) 32 ln(|2t 1|) 32
12t 1
= 2 ln(|x +x2 x + 1|) 32 ln(|2x 1 + 2
x2 x + 1|)
32 ln(1
2x 1 + 2x2 x + 1) + C.
-
Riemann
.
JJ II
J I
183 pi 273
1.2.35
-
Riemann
.
JJ II
J I
184 pi 273
pi : pi pi pi . pi Pn pi.. U (f, Pn) .
1n
nk=1
nek =
1n
ne +
1n
ne2 + + 1
nnen
=1ne
1n +
1ne
2n + + 1
ne
nn
[0,1], Pn = {x0 = 0, x1 = 1n , x1 = 1n , , xn = 1} f : R f (x) = ex . 10 exdx = [ex ]10 = e e0 = e 1.pi U (f, Pn). U (f, Pn) = 1n
nk=1
nek limn U (f, Pn) = limn 1n
nk=1
nek
e 1 = limn 1nn
k=1nek.
2.2.1
-
Riemann
.
JJ II
J I
185 pi 273
pi : Pn limn[U (f, Pn) L(g, Pn)] = 0. f R([a, b]) pi Pn
bafdx = limn L(f, Pn).
pi P n bahdx = limn U (h, Pn). Qn = Pn P n
( Qn pi Pn ) pi
L(f, Pn) L(f, Qn) L(g, Qn) U (g, Qn) U (h,Qn) U (h, P n),pi f (t) g(t) h(t). 0 U (g, Qn)L(f, Qn) U (h, P n)L(f, Pn) limn[U (h,Qn)L(f, Qn)] =
bahdx b
afdx = 0. pi
limn[U (g, Qn) L(g, Qn)] = 0
g R([a, b]) bagdx limn U (g, Qn) =
bafdx.
2.2.2
-
Riemann
.
JJ II
J I
186 pi 273
pi : f , 0 ba|f (t)|dt pi .
f , 0 pi t0 [a, b] : |f (t0)| = , 0. pi f pi pi U t0 |f (t)| > 2 , t U . f , 0 ba|f (t)|dt
U|f (t)|dt 2 U > 0.
2.2.3
-
Riemann
.
JJ II
J I
187 pi 273
pi : Pn f (x) = 1x U ( 1x , Pn) = 1 +
12 + 1n1 L( 1x , Pn) = 12 + 1n .
Pn = {1 < 2 < ... < n} [1, n].
L(f, Pn) =n
k=2mk(tk tk1) =
nk=2
mk =n
k=2
1k,
U (f, Pn) =n
k=2Mk(tk tk1) =
nk=2
Mk =n
k=2
1k 1 .
2.2.4
-
Riemann
.
JJ II
J I
188 pi 273
pi : sin x [a, b] -. Pn = {a, a+h, a+2h, a+nh = b}, h = ban . Riemann, L(sin x, Pn). limn L(sin x, Pn) =
ba
sin xdx,
L(sin x, Pn) =n
k=1sin(a + (k 1)h)h = h
nk=1
sin(a + (k 1)h) = h cos(a h2 ) cos(b h2 )2 sin( h2 )
.
limh0 h2 sin( h2 ) = 1, limn L(sin x, Pn) = ba
sin xdx = cos b cosa.
2.2.5
-
Riemann
.
JJ II
J I
189 pi 273
pi : [a, b] -. pipi ) m , 1, ) m = 1.) Pn = {a, ah, ah2, ahn}, h = n
ba . h 1
n . Riemann Pn.
Sn(f, Pn) = (ah a)am + (ah2 ah)amhm + + (ahn ahn1)amh(n1)m= am+1(h 1) + am+1hm+1(h 1) + + am+1h(n1)(m1)(h1)
= am+1(h 1){1 + hm+1 + (hm+1)n1}= am+1(h 1) (h
m+1)n 1hm+1 1 {1 + h
m+1 + (hm+1)n1} = h 1hm+1 1 (b
m+1 am+1).
limn1 h1hm+11 =1
m+1 . baxmdx = limn Sn(f, Pn) = b
m+1am+1m+1 .
) m = 1 m + 1 = 0 Sn(f, Pn) = n(h 1). pi ahn = b n = ln blnalnh , bax1dx = lim
nSn(f, Pn) = limh1h 1lnh
(ln b lna).
2.2.6
-
Riemann
.
JJ II
J I
190 pi 273
pi :
Pn = {a, a + b an
, a +2(b a)
n, , a + n(b a)
n= b}.
||Pn || = ban 0. Riemann Pn
S(Pn , f,) =n
k=1f(a + k
b an
)||Pn ||.
limnS(Pn , f,) = limn
nk=1
f(a + k
b an
)||Pn || =
baf (x)dx.
2.2.7
-
Riemann
.
JJ II
J I
191 pi 273
pi : limn bann
k=1 f(a + k ban
)=
baf (x)dx. a = 0,
b = 1
limn
1n
ni=1
f (i
n).
2.2.8
-
Riemann
.
JJ II
J I
192 pi 273
pi : an
an =1n{ 11 + 1n
+ + 11 + nn}.
pipi pi f (x) = 1x+1 x =1n , , nn . n ,1
n 0 nn 1. [0,1] pi f pi . Pn = {0, 1n , 2n , , nn = 1} pi limn ban
nk=1 f
(a + k ban
)=
baf (x)dx,
limnan = limn{
1n{ 11 + 1n
+ + 11 + nn}} =
10
11 + x dx
2.2.9
-
Riemann
.
JJ II
J I
193 pi 273
pi : f (x) = ex R [0,1].
Pn = {x0 = 0, x1 = h, x2 = 2h, , xn = nh = 1},pi h = 1n . = {0, h,2h, , (n 1)h} 1
0exdx = lim
n
h n1k=0
f (kh)
= limnhe0 n1
k=0eh
k
= lim
n
(heh
n 1eh 1
)= (e 1) lim
nh
eh 1 = (e 1).
2.2.10
-
Riemann
.
JJ II
J I
194 pi 273
pi : C 0 < C < f (c). pi f pi > 0 x [c , c + ] [a, b] f (c) > C. pi b
af (x)dx > 0 =
ca
f (x)dx + c+c
f (x)dx + bc+
f (x)dx 0 + 2C + 0 > 0.
c = a ( c = b) pi [a, a + ] ( [b , b]).
2.2.11
-
Riemann
.
JJ II
J I
195 pi 273
pi :) 1
0x2dx =
x3
3 |10 =
13
03 =
13
) 10x3dx =
x4
4 |10 =
14
04 =
14
) pi0
cos(x)dx = sin(x)|pi0 = sin(pi) sin(0) = 1 0 = 1.
) 11
(2x2 x3)dx = 2 11x2dx
11x3dx = 2x
3
3 |11
x4
4 |11 = 2(
13
13 ) (
14
14 ) =
43
) 32ex/2dx = 2ex/2|32 = 2(e3/2 e1) =
2e(1 e1/2).
2.2.12
-
Riemann
.
JJ II
J I
196 pi 273
pi : f (x) = x + pi g(x) = sin(x) x [0, pi]. f g pi g(x) 0 [0, pi]. pi pi pi pi [0, pi] pi
0(x + pi) sin(x)dx = ( + pi)
pi0
sin(x)dx
2.2.13
-
Riemann
.
JJ II
J I
197 pi 273
pi : f (x) = 1x g(x) =sin(x) x [a, b]. pi [a, b] b
a
sin(x)x
dx =1a
a
sin(x)dx +1
b
sin(x)dx.
pi a
sin(x)dx = cos(a) cos( ), b
sin(x)dx = cos( ) cos(b) | a
sin(x)dx | 2,| b
sin(x)dx | 2,
| ba
sin(x)x
dx | 2b+2a 2a+2a=4a.
2.2.14
-
Riemann
.
JJ II
J I
198 pi 273
pi : , f (x) = g(x) x3 x2 x + 1 = x + 1. (1,0), (0,1), (2,3). g(x) f (x) 1 x 0 f (x) g(x) 0 x 2.
E =
21|f (x)g(x)|dx =
01
[(x3x2x+1)(x+1)]dx+ 20
[(x+2)(x3x2x+1)]dx = 3712
2.2.15
-
Riemann
.
JJ II
J I
199 pi 273
pi : pi 2pi0
1 sin(x)dx.
E =
2pi0
(1 sin(x))dx =
2pi0
1 + cos(x + pi2 )dx,
t = x + pi2
E =
5pi4
pi2
1 + cos(2t)dt = 2
2 5pi
4
pi2
| cos(t)|dt = 22[ pi
2
pi4
cos(t)dt 5pi
4
pi2
cos(t)dt] = 4(2).
2.2.16
-
Riemann
.
JJ II
J I
200 pi 273
pi : pi y = x2, x + y = 2, x2 = 2 x x = 1 x = 4. (1,1) (4,2). 0 x 1 pi pi 1 x 4 pi pi.
E =
10
[x (x)]dx +
41
[(2 x) (x)]dx = 92 . pi pi y
E =
12
[(2 y) y2]dy = 92 .
2.2.17
-
Riemann
.
JJ II
J I
201 pi 273
pi : 0 pi pi pi (3, 2pi3 ). (3, 2pi3 ) , pi (0,0) (0, pi2 ) .
E = 2 2pi3
0(2 2 cos())2d 12
2pi3
pi2
(6 cos())2d
=
2pi3
0(6 8 cos() + 2 cos(2))d
2pi3
pi2
(1 + cos(2))d
= [6 8 sin() + sin(2)]2pi/30 18[ +sin(2)
2 ]2pi/3pi/2 = pi.
2.2.18
-
Riemann
.
JJ II
J I
202 pi 273
pi : pi y = x2 pi x = t2,y = 2t, t [0,1]
S =
10
(2t)2 + 4dt = 2
t2 + 1dt
= 2[12 tt2 + 1 + 12 ln |t +
t2 + 1|]10 =
2 + ln(1 +
2).
2.2.19
-
Riemann
.
JJ II
J I
203 pi 273
pi :
S =
2pi0
x (t)2 + y(t)2dt =
2pi0
( sin(t) sin(2t))2 + (cos(t) + cos(2t))2dt
=
2pi0
2(1 + cos(t))dt = 2
2pi0
cos2(
t
2 )dt
= 2 2pi0
| cos( t2 )|dt = 2 pi0
cos(t
2 )dt 2 2pipi
cos(t
2 )dt = 8
2.2.20
-
Riemann
.
JJ II
J I
204 pi 273
pi : pi limn bann
k=1 f(a + k ban
)=
baf (x)dx. -
41
(2x3 5x)dx = limn
3n
ni=1
f(1 + 3i
n
)= lim
n3n
ni=1
[2(1 + 3i
n
)3 5
(1 + 3i
n
)]
= limn
3n
ni=1
[3 + 3 i
n+ 54 i
2
n2+ 54 i
3
n3
]
= limn
3nni=1
(3) + 9n2
ni=1
i +162n3
ni=1
i2 +162n4
ni=1
i3
= limn
3n (3)n + 9n2 n(n + 1)2 + 162n3 n(n + 1)(2n + 1)6 + 162n4(n(n + 1)
2
)2= lim
n
{9 + 921(1 +
1n) + 27(1 + 1
n)(2 + 1
n) +
812 (1 +
1n)2
}= 9 + 92 + 27 2 +
812 = 90
2.2.21
-
Riemann
.
JJ II
J I
205 pi 273
pi : u = 3x + 4 du = 3dx. pi x = 0, u = 4 x = 4,u = 16. 4
0
3x + 4dx = 13
146udu =
1323 [u
3/2]146
=29 (16
3/2 43/2) = 1129 .
2.2.22
-
Riemann
.
JJ II
J I
206 pi 273
pi : pi y = sin(x) y = cos(x) [0, pi/2] pi/4,
(2)/2. cos(x) sin(x) 0 x pi/4 cos(x) sin(x)
pi/4 x pi/2. pi
E =
pi/20
| cos(x) sin(x)|dx
=
pi/40
(cos(x) sin(x))dx + pi/2pi/4
(sin(x) cos(x))dx
= [sin(x) + cos(x)]pi/40 + [ cos(x) sin(x)]pi/2pi/4= (
12+
12 0 1) + (0 1 + 1
2+
12) = 2
2 2.
2.2.23
-
Riemann
.
JJ II
J I
207 pi 273
pi : pi pi pi pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.
e = a2b2a2 < 1 e cos(t) = u
E = 2piab pi0
sin(1 e2 cos2(t))dt = 2pi ab
e
ee
1 u2du
= 2piab[ arcsin(e)e
+1 e2].
2.2.24
-
Riemann
.
JJ II
J I
208 pi 273
pi : pi pi pi pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.
E = 2pi 2pi0
a2(1 cos(t))2(1 cos(t))dt= 4pia2
2pi0
(1 cos(t)) sin( t2 )dt =643 pia
2.
2.2.25
-
Riemann
.
JJ II
J I
209 pi 273
pi : pi pi pi pi E = 2pi
ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.
pi pipi l, m pipi pi x . y = 0 pi t = 0 t = 3. x = 0 x = 3. pi x- (0,0) (3,0). pi pi pi x - .
E = 2pi 30
|y(t)|x (t) + y(t)dt= 2pi
30
t3 (t2 3)(4t2 + (t2 1)2)dt
= 2pi 30
t3 (t2 3)(1 + t2)dt = 3pi.
2.2.26
-
Riemann
.
JJ II
J I
210 pi 273
pi : pi pi pi f , g, f (x) g(x) 0, x [a, b] V = pi b
a{f (x2) g(x)2}dx. pipi
pi
V = pi
21{(x + 2)2 x4}dx = 72pi5
2.2.27
-
Riemann
.
JJ II
J I
211 pi 273
pi : pi pi pi - f , x [a, b] V = pi b
af (x)2dx.
V = pi
aa
y2dx = pi
aa
b2(1 x
2
a2
)dx =
43piab
2
2.2.28
-
Riemann
.
JJ II
J I
212 pi 273
pi : pi pi pi - x = g(t), y = f (t), t [t1, t2] V = pi
t2t1f (t)2g(t)dt.
V = 2pi a0
y2dx
x(t1) = 0 t = pi2 x(t2) = a t = 0
V = 2pi 2pi0
y2dx = 2pi 0pi/2
a2 sin6(t)(3a cos2(t) sin(t))
= 6pia3[ pi/20
sin7(t)dt pi/20
sin9(t)dt]
= 6pia3(674523 +
89674523
)=
32105pia
3.
2.2.29
-
Riemann
.
JJ II
J I
213 pi 273
pi : [0,1] 10 . xi = i10 , i = 1, ,10 : f (0) = 1, f (x1) = 1.00005, f (x2) = 1.00080, f (x3) = 1.00404, f (x4) =1.01272, f (x5) = 1.03078, f (x6) = 1.026283, f (x7) = 1.11360, f (x8) = 1.18727, f (x9) =1.28690, f (x10) = 1.41421.
T10 =120 [f (0) + 2f (x1) + + 2f (9) + f (1)] = 1.09061
.
2.2.30
-
Riemann
.
JJ II
J I
214 pi 273
pi : [0,1] 10 . xi = i10 , i = 1, ,10 : f (0) = 1, f (x1) = 1.00005, f (x2) = 1.00080, f (x3) = 1.00404, f (x4) =1.01272, f (x5) = 1.03078, f (x6) = 1.026283, f (x7) = 1.11360, f (x8) = 1.18727, f (x9) =1.28690, f (x10) = 1.41421.
S =130 [f (0) + {f (
110 ) + f (
310 ) + f (
510 ) + f (
710 ) + f (
910 )}
+{f ( 210 ) + f (410 ) + f (
610 ) + f (
810 )} + f (1)]
=32.68473
30 = 1.08949.
2.2.31
-
Riemann
.
JJ II
J I
215 pi 273
pi : nn+3 < 1 n
2n (n+3) 1 3 > 1
n=1
n+2n3 .
3.2.5
-
Riemann
.
JJ II
J I
220 pi 273
pi : . an = 3n
n10 , an , 0 N.limn | an+1an | = limn |
3n+1(n+1)10
3nn10
| = limn 3n10(n+1)10 = 3 limn n10
(n+1)10 = 3 limn1
( n+1n )10 =
3 limn 1(1+ 1n )10 = 3 > 1. n=1 3nn10 pi.
3.2.6
-
Riemann
.
JJ II
J I
221 pi 273
pi : . an = n5n/2 0 limn |an |1n =
limn a1nn = limn
(n
5n/2) 1n= limn n
1n
512= 1
5limn n1/n = 15 < 1.
n=1
n5n/2
.
3.2.7
-
Riemann
.
JJ II
J I
222 pi 273
pi : . an = 9n
n! , 0 | an+1an | = |9n+1(n+1)!9nn!| =
| 9n+1n!9n (n+1)! | = | 9n+1 | = 9n+1 0 n N. pi pi n=1
9nn! .
3.2.8
-
Riemann
.
JJ II
J I
223 pi 273
pi : . an = nn
(3n+1)n limn |an |1n =
limn a1nn = limn
(nnn
n(3n+1)n)= limn |n||3n+1| = limn
n3n+1 = limn
13n+1n
= limn 13+ 1n=
13 . pi pi
n=1
nn
(3n+1)n .
3.2.9
-
Riemann
.
JJ II
J I
224 pi 273
pi : . an = 1+cos2(nx)
2n |an |1n = a
1nn =
n1+cos2(nx)
2 . 12 n
an 12 n
2, pi pi pi pi
pi n1 + cos2(nx) n
2. pi
nan 12 .
n=11+cos2(nx)
2n .
3.2.10
-
Riemann
.
JJ II
J I
225 pi 273
pi : n=1 (n!)nnn2 an = (n!)nnn2 . n=1 an . an > 0 n N |an | = an. .limn |an |1/n = limn
n(n!)nnnnn = limn
|n!|nn = limn
n!nn = p R
{,+}. 0 p < 1 n=1 an an = (n!)nnn2 0 n .pi pipi p = 0 limn n!nn = p = 0. bn = n!nn .
n=1 bn pi limn bn = 0.
. bn , 0, n N.bn+1bn
= |(n+1)!
(n+1)n+1n!nn
| = | nn (n+1)n!(n+1)n+1n! | = nn
(n+1)n =(
11+ 1n
)n= 1(1+ 1n )n
1e < 1 n .
n=1 an
limn (n!)n
nn2= 0.
3.2.11
-
Riemann
.
JJ II
J I
226 pi 273
pi : n=1 12n1 , pi Sn = 1 +
12 +
122 +
123 + +
12n 1 =
1 ( 12 )n1 ( 12 )n
112= 2 n ,
n pi 12 .
n=11
2n1 = 2. pi0 11+2n1 12n1 . pi pi
n=1
11+2n1 .
3.2.12
-
Riemann
.
JJ II
J I
227 pi 273
pi : n=1 1n , ( ) pi n=1
1n = +
0 1n 2n1n pi
n=1
2n1n
pi.
3.2.13
-
Riemann
.
JJ II
J I
228 pi 273
pi : bn = 12n1 an =1n .
n=1
1n
pi.
limn
anbn
= limn
1n1
2n1= lim
n(2 1n) = 2.
n=1 12n1 pi.
3.2.14
-
Riemann
.
JJ II
J I
229 pi 273
pi : a 0 + 1na , 0. a = 1 +. a > 0 f (x) = 1xa [1,+) a , 1
limnF (n) = limn
n1
tadt = limn[
11 a n
1a 11 a ] ={ a < 1
11a a > 1
pi , , a > 1.pi pi pipi n=1 1na 1a1 + 1 = aa1 .
3.2.15
-
Riemann
.
JJ II
J I
230 pi 273
pi : an = n2n .
n|an | = n
| n2n | =
nn
2 12 n .
n=1 n2n . 3.2.16
-
Riemann
.
JJ II
J I
231 pi 273
pi : an = nn
n! , 0. .
|an+1an
| = |(n+1)n(n+1)!nn
n!
| = |n!(n + 1)n+1
n!(n + 1)nn | = |(n + 1)n
nn| =
(n + 1n
)n=
(1 + 1
n
)n e > 1,
n . pi n=1 nnn! pi.
3.2.17
-
Riemann
.
JJ II
J I
232 pi 273
pi : an =(1 + 1n
)n2. .
|an | 1n = a1nn =
n
(1 + 1
n
)n2=
(1 + 1
n
)n e > 1,
n . pi n=1 (1 + 1n )n2 pi.
3.2.18
-
Riemann
.
JJ II
J I
233 pi 273
pi : 3n 2n 13n 12n . 12n 0 12n + 13n 22n , n N.pi n=1 22n .
n
| 22n | =
n
22n =
n22
12 < 1 n .
n=1 ( 12n + 13n ) . piSn =
(12 +
122 +
123 + +
12n
)+
(13 +
132 +
133 + +
13n
)=
12
( 12 )n 1
12 1
+13
( 13 )n 1
13 1
32 n .
3.2.19
-
Riemann
.
JJ II
J I
234 pi 273
pi : n=1 n1357(2n1) . an =n
1357(2n1) , 0. .
|an+1an
| = an+1an
=
n+11357(2n+1)
n1357(2n1)
=n + 1
n(2n + 1) 0 n .
n=1 n1357(2n1) limn n1357(2n1) = 0.
3.2.20
-
Riemann
.
JJ II
J I
235 pi 273
pi : 0 an 9 0 an10n 910n . n=1 910n = 9n=1 110n = 9 19 = 1, pi = 110 < 1.pi pi n=1 an10n pi [0,1].
3.2.21
-
Riemann
.
JJ II
J I
236 pi 273
pi : an = 5n
7nn32, 0 n N .
|an+1an
| =5n+1
7n+1(n+1)32
5n
7n (n)32
=5n+17nn 32
5n7n+1(n + 1) 32=
5n 327(n + 1) 32
=
57
( nn + 1
) 32=57
1n+1n
32 = 57 11 + 1n
32 57 < 1 n .
3.2.22
-
Riemann
.
JJ II
J I
237 pi 273
pi : an = sin( 1n ).
n=11n bn =
1n . -
. bn > 0, an n N anbn =sin( 1n )
1n
1 > 0.pi n=1 1n = + pi pi
n=1 sin(1n ) pi.
3.2.23
-
Riemann
.
JJ II
J I
238 pi 273
pi : f (x) = 1x(ln(x))a , a > 0 x [2,+) pi pipi- . a , 1
limnF (n) = limn
n2
1t(ln(t))a
dt =1
1 a limn[ln(n)1a ln(2)1a]
=
{ 1a1 (ln(2))
1a , a > 1+, a < 1.
a = 1 limn F (n) = limn[ln(ln(n)) ln(ln(2))] = +.
3.2.24
-
Riemann
.
JJ II
J I
239 pi 273
pi : n=1(1)n n2+12n3+n1 . an = n
2+12n3+n1 . an an 0 n . pi .
pi pi , n=1 n2+12n3+n1 . pi
n2 + 12n3 + n 1
n2
2n3 + n 1 2n2
2n3 + 2n3 + 2n3 =14n .
n=1 n2+12n3+n1 pi n=1(1)n n2+12n3+n1 -.
3.2.25
-
Riemann
.
JJ II
J I
240 pi 273
pi : n=1 (1)n+12n . an =
12n . an =
12n an 0 n . pi .
pi pi , n=1 12n . r = 1/2 < 1 pi.
3.2.26
-
Riemann
.
JJ II
J I
241 pi 273
pi : Sn =n
k=12+(1)k
2k , Sn =nk=1
22k +
nk=1
(1)k2k . limn
nk=1
12k1 =
11 12
= 2. -
Tn =n
k=1(1)k2k
Tn =n
k=1(12 )
k = 12 + (12 )
2 + + (12 )n
= 12 [1 + (12 ) + + (
12 )
n1] = 12( 12 )n 1 12 1
.
Tn =( 12 )1
3 , limn Tn =13 limn( 12 ) 13 = 13
pin=1
2 + (1)n2n =
n=1
22n +
n=1
(1)n2n = 2
13 =
53 .
3.2.27
-
Riemann
.
JJ II
J I
242 pi 273
pi : limn 1n = 0, limn ln(n+1n ) = ln(1) =
0, 1n > ln(n+1n ) >
1n+1 . ( c Euler,
c = 0.577215).
3.2.28
-
Riemann
.
JJ II
J I
243 pi 273
pi : s > a k , 1
lims
sa
dx
xk=
11 k lims[s
1k a1k] ={ 1
1ka1k k > 1
+ k < 1. k > 1 pi k < 1. k = 1
lims
sa
dx
x= lim
s[ln(s) ln(a)] = +.
4.2.1
-
Riemann
.
JJ II
J I
244 pi 273
pi : 0
11 + x2dx = limt
0t
11 + x2dx = limt arctan(t) =
pi
2 .
4.2.2
-
Riemann
.
JJ II
J I
245 pi 273
pi :
11 + x2dx =
0
11 + x2dx +
0
11 + x2dx
= limt
0t
11 + x2dx + lims
s0
11 + x2dx
= limt arctan(t) + lims arctan(s) = (
pi
2 ) +pi
2 = pi.
4.2.3
-
Riemann
.
JJ II
J I
246 pi 273
pi : )
limt
t1
x1 + x2
dx = limt[
1 + t2 2] = +.
.)
limt
t0
cos(t) = limt sin(t).
limt sin(t) pi. .)
limt
t0exdx = lim
t[1 et] = 1.
.)
limt
t1
ln(x)x
dx = limt(
12 ln(t)
2) = +. .
4.2.4
-
Riemann
.
JJ II
J I
247 pi 273
pi : ex2 [0,+]. 1 e
x2dx t [1,+] et2 < et F (t) = t1 ex2dx t1 e
xdx 1 e
xdx
limt
t0exdx = lim
t[1 et] = 1.
1 e
x2dx . pi pi 0
ex2dx =
10ex
2dx +
1
ex2dx,
10 e
x2dx pi 0 e
x2dx .
4.2.5
-
Riemann
.
JJ II
J I
248 pi 273
pi : | sin(x)|xa 1xa | cos(x)|xa 1xa x [a, b]. -
1
1xa dx a > 1
1
sin(x)xa dx
1
cos(x)xa dx
pi a > 1.pi t > 1 t
1
sin(x)xa
dx =
[cos(x)
xa
]t1 a
t1
cos(x)x1+a
dx.
t1
cos(x)x1+a dx 1 + a > 1 a > 0 pi
1
sin(x)xa
dx = limt
[cos(x)
xa
]t1 a lim
t
t1
cos(x)x1+a
dx = cos(1) a 1
cos(x)x1+a
.
pi a > 0 .
4.2.6
-
Riemann
.
JJ II
J I
249 pi 273
pi : pi t3
dxx2+x2 , t > 3 t
3
dx
x2 + x 2 = 13
t3
dx
x + 2 +13
t3
dx
x 1 =13 ln(
52t 1t + 2 ).
t 3
dx
x2 + x 2 =13 ln(
52 ).
4.2.7
-
Riemann
.
JJ II
J I
250 pi 273
pi : pi 1
x
(1 + x2)dx = 1
xd
1(1 + x2)
= limx
( x
1 + x 12
)+
1
dx
2x(1 + x)
=12 +
12
1
dx
2x(1 + x)
=12 +
12
1
2tdtt(1 + t2)
=12 + limx(arctan
(x) pi4 )
=12 +
pi
2 pi
4 =12 +
pi
4 .
4.2.8
-
Riemann
.
JJ II
J I
251 pi 273
pi : Cauchy a < x1 < x2 x2x1
sin(x)x
dx = x2x1
1xd cos(x) =
cos(x1)x1
cos(x2)x2
x2x1
cos(x)x2
dx.
pi
| x2x1
sin(x)x
dx | 1x1
+1x2
+ | x2x1
cos(x)x2
dx |
1x1
+1x2
+
x2x1
1x2
dx =2x1
< . (4.4)
pi = 2 pi Cauchy a
sin(x)x dx
.
4.2.9
-
Riemann
.
JJ II
J I
252 pi 273
pi : f (x) = 1x4+1
0 < f (x) = 11 + x4 1) pi
a
11+x4dx .
4.2.10
-
Riemann
.
JJ II
J I
253 pi 273
pi : f (x) = 1(1+x3)1/3
f (x) =1
(1 + x3)1/3>
11 + x = g(x),
x [0,+]. a
11+x dx = + pi
a
1(1+x3)1/3dx = +
4.2.11
-
Riemann
.
JJ II
J I
254 pi 273
pi : f (x) = x22x4x2+1 g(x) =x2
x4 =1x2 (pi pi
f pi). pi limx f (x)g(x) =12
1
dxx2 < + pi
0
x2
2x4x2+1dx < +
4.2.12
-
Riemann
.
JJ II
J I
255 pi 273
pi : Cauchy > 0 x1, x2 > | x2
x1x sin(x4)dx | < > 0. 0 x sin(x4)dx = 10 x sin(x4)dx +
1 x sin(x4)dx x2
x1
x sin(x4)dx =14
x42x41
sin(t)tdt
= 14cos(t)
t|x42x41 18
x42x41
cos(t)t3/2
dt
= 14[cos(x22 )x22
cos(x21 )
x21
] 18
x42x41
cos(t)t3/2
dt.
| x2x1
x sin(x4)dx | 14 (1x22
+1x21
) +18
x42x41
dt
t3/2
=14 (
1x22
+1x21
) 14 (1x22
1x21
) =12x21
< . (4.5)
= 12 pi Cauchy pi 0 x sin(x
4)dx,.
4.2.13
-
Riemann
.
JJ II
J I
256 pi 273
pi : 1(xa)k (a, b]. a < t < b k , 1 b
t
dx
(x a)k =1
1 k[
1(b a)k1
1(t a)k1
],
pilimta
bt
dx
(x a)k ={ 1
1k (b a)1k k < 1+ k > 1.
ba
dx(xa)k k < 1 pi k > 1. k = 1
limta bt
dxxa = limta[ln |b a| ln |t a|] = , pi.
4.2.14
-
Riemann
.
JJ II
J I
257 pi 273
pi : pi ca
dx(xa)(bx) ,
bc
dx(xa)(bx) .
pi
limxa
(x a)1/2(x a)(b x) =
1(b a) ,
limxb
(b x)1/2(x a)(b x) =
1(b a) .
pi x = a cos2(t) + b sin2(t) ba
dx(x a)(b x) = 2
pi/20
dt = pi.
4.2.15
-
Riemann
.
JJ II
J I
258 pi 273
pi : x = 1t 10
sin(1x)dx =
l
sin(t)t2
dt.
l
sin(t)t2 dt pi
| l
sin(t)t2
dt | l
1t2dt < +.
pi 10 sin(
1x )dx .
4.2.16
-
Riemann
.
JJ II
J I
259 pi 273
pi : pi0
dx
sinp(x)=
pi/20
dx
sinp(x)+
pipi/2
dx
sinp(x).
pi x = pi t pipi/2
dxsinp(x)
pi/20
dxsinp(x) . pi limx0+ x
p 1sinp(x) = 1 pi pi,pi
, 0 < p < 1.
4.2.17
-
Riemann
.
JJ II
J I
260 pi 273
pi :
(x) = 10tx1etdt +
+1
tx1etdt.
pipi ) x 1 )x < 1 ) x 1 : pipi 10 tx1etdt
+1 t
x1etdt
limt
tx1et1t2
= limt
tx+1
et= 0,
1
1t2dt < . pi
+1 t
x1etdt (x) x 1) x > 1 : pipi
+1 t
x1etdt pi pipi ).
10 t
x1etdt . t = 1s 1
0tx1etdt =
+1
ts1e1/sds.
pi
lims
ts1e1/s
sx1= 1,
1 s
x1ds < x > 0 pi
10 t
x1etdt (x) x > 1.
4.2.18
-
Riemann
.
JJ II
J I
261 pi 273
pi :
B(x, y) = 10tx1(1 t)y1dt =
c0tx1(1 t)y1dt +
1ctx1(1 t)y1dt,
0 < c < 1.pi f (t) = tx1(1 t)y1
limt t
1x f (t) = limt(1 t)
y1 = 1,
limt(1 t)
1yf (t) = limt t
x1 = 1.
pi pi c0 t
x1(1t)y1dt 0 < 1 x < 1 1
ctx1(1 t)y1dt 0 3. pi pi x = 1 x = 3. x = 1 n=1 n+1n2+1 (1)n, pi pi ( an = n+1n2+1 ) . x = 3 n=1 n+1n2+1 pipi.pi n=1 n+1n2+1 (x 2)n [1,3) R = 312 = 1.
4.2.22
-
Riemann
.
JJ II
J I
265 pi 273
pi :
limn
n
3n2n+4 |x
n | = 32 |x |.
pi |x | < 23 23 < x < 23 pi x > 23 x < 23 . pi pipi pi x = 23 x = 23 . x = 23
n=1
18 (1)n. pi pi-
. x = 23
n=1
18 . pi pi.
pi n=1 3n2n+4 xn ( 23 , 23 ) R = 23 .
4.2.23
-
Riemann
.
JJ II
J I
266 pi 273
pi :
limn
xn+1
(n+1)!xn
n!
= limn
x
n + 1 = 0,
x , 0 n=1 xnn! = 0 x = 0. x R (,+) R = +.
4.2.24
-
Riemann
.
JJ II
J I
267 pi 273
pi : x , 1
limn
nnn(x 1)n = lim
nn|x 1| = +.
x = 1 n=1 nn(x 1)n = 0 . pi - [1,1] R = 0.
4.2.25
-
Riemann
.
JJ II
J I
268 pi 273
pi : fn(x) = (1)n+1 xnn x [0, a] a < 1,
|fn(x)| = |(1)n+1 xn
n| = x
n
n a
n
n.
n=1 ann 0 a < 1 pi x (1,1) R = 1.
4.2.26
-
Riemann
.
JJ II
J I
269 pi 273
pi : fn(x) = xn ln(x)n x [0,1] . -
x [0,1] |fn(x)| = fn(x) fn(x) xm = 1ne fn(xm) = 1n2e
|fn(x)| = xn ln(x)n
1n2e
.
n=1 1n2e pi x [0,1].
4.2.27
-
Riemann
.
JJ II
J I
270 pi 273
pi :
limn
nanxn = lim
na|x | = a|x |.
pi pipi a|x | < 1 |x | < 1a . - ( 1a , 1a ) R = 1a .
4.2.28
-
Riemann
.
JJ II
J I
271 pi 273
pi : f : (R, R) R pi f (x) = n=0 anxn. f (x) = n=0 nanxn. pi pi
2f (x) + f (x) = 0 2e2x f (x) + e2x f (x) = 0 (e2x )f (x) + e2x f (x) = 0 (e2x f (x)) = 0 e2x f (x) = c f (x) = ce2x .
ex = n=0 xnn! e2x = n=0 (2x)nn! f (x) = cn=0 (2)nn! xn. pin=0
an =n=0
(c(2)nn!
)xn.
pi an = c (2)n
n! .
4.2.29
-
Riemann
.
JJ II
J I
272 pi 273
pi : n=0 anxn R, |x | < R. pi n=0 anx2n = n=0 an(x2)n pi |x2| < R |x | < R. n=0 anx2n
R.
4.2.30
-
Riemann
.
JJ II
J I
273 pi 273
pi : pi
limn
(n + 1)axn+1
naxn= lim
n((n + 1)
n)ax = x.
R = 1. x = 1 n=1 na(1)n pi a < 0 . x = 1 n=1 na pi a < 1.pi a < 1 [1,1], 1 a < 0 [1,1) a 0 (1,1).
4.2.31
A'oristo Olokl'hrwmaStoiqe'ia Jewr'iacAsk'hseic
Olokl'hrwma RiemannStoiqe'ia Jewr'iacAsk'hseic
Seir'ecStoiqe'ia Jewr'iacAsk'hseic
Genikeum'ena oloklhr'wmataStoiqe'ia Jewr'iacAsk'hseic