文档内容
xsin x + sin x
例3.14 计算定积分 dx
0 3 + sin2 x
= / XSmYNx +Sm
T
/ dosx
1)/sux
( ( 1)
+ xdx
= +
=
- .
22 105 X
-
-E
= Ellis
+
=1
例3.15 设 f (x) = arctan(x − 1)2 ,且 f (0) = 0,计算 f (x)dx.
0
fixl t
fix
to arcem cel
arcem
(+
= at
=
·
X fl (oxfinax fil-lxarcamix Max
Cax =
35- : = - -
CavimE-Pax-lox +
= arcem(x ax
/
: )arcem(x-1)" >1
(x-1)"
- (x+ ax-1) 11
= = arih d(x
-
-
-
W
2
t (x 1)
= - Carseme
- If
at
. = arra i ttanent
=
= =
amata
(0'(x-1)
11fiax fix . fix
Pro Fi (x-1) ex
: = -
↑
arcemix-1ax
%
- cx+)
=
·
So area c
%Claremstaelax Coax
+
faux
35 at
= = .
==
↑
Sjat] Carlem (t-(1
arem(tax
t)at t · (11)
= = -
t X
=
S
(x-15ax 1)
-J
(x-1) accen
= .
3
!
G
X = 1 1 1
例3.16 设a 1, f (x), g(x)在 ,a 上连续,如果x ,a ,都有
a a
1 1
f (x) = f , g(x) + g = A，
x x
A 为常数，证明：
f (x)g(x) A f (x) f (x)
a a a
dx = dx = A dx.
1 1
x 2 x 1 x
a a
IfNgHax(fl
LEDE
:
I at
X
= E
-tat
ax
=
fit (A
- -914) I ax - I fin
at
=
t
Ifax
=
=>
a2 ax
Lax
=
I S +
x
=
&x
1
)
=
19 x =
:
- Al ?
=
-
-=:/a
35 =
(
F
- F(a)
=
,
flat
f(ul 9(a)
9) (fal-fat
Fal - (-)
.
=
. -
-
t
a
fal-qual
fal(A-9(a)) ) f
*
=
+
t -
a a
Aquafaga-
=
Fal #PF(l
: ( Fal
= = 0 : = O
.例3.17 设 f (x)在区间[−a,a](a 0)上具有二阶连续导数, f (0) = 0,
(1)写出 f (x)的帯拉格朗日余项的一阶麦克劳林公式;
f 505X2(R)
fa fiax
f(x)
= + +
.
+ -
fia
x
=
.a
(2)证明在[−a,a]上至少存在一点,使a3 f () = 3 f (x)dx.
−a
[fax
fil 3 f
[i] : 2] TE
=
,
a
FIRE
E
LENA a a)
: -
.
fix a] HEm , IM
Ea
:
.
+.
fix fiax
#
(1)
=
:
IfMax (a 1 .
fix.xdx
: x
= +
!fix
= o +
max =L = fax
fax
:a
(2)证明在[−a,a]上至少存在一点,使a3 f () = 3 f (x)dx.
−a
max =L
fax
= fax
:
3
~
fax
3) fax
:m
M
.
·
-
as
>(
finax
-
ITE EYEEa a) fMl
: =
.
.
93例3.18 (1)证明柯西-施瓦茨不等式：设 f (x), g(x)在 [ a , b ] 上连续, 则
2
b b b
f (x)g(x)dx f 2(x)dx g2(x)dx.# * F' FIR
:
a a a
A.f
A : I X
a
I fill *
Dafigael" at get
=
(l figual" (fine" gits
* FM at
-
=
Fix 2 fitig fit fix Sin -Ifitat giv
de 9-
= . . . . at
·
To figi fix MdE In"fix gitt de - fa
= . fact gix at
-
( [fingin fin git-2
fitge fin
= - + - 9 ) at
. ,例3.18 (1)证明柯西-施瓦茨不等式：设 f (x), g(x)在 [ a , b ] 上连续, 则
2
b b b
f (x)g(x)dx f 2(x)dx g2(x)dx.
a a a
(a* [fgt-fgrd]"at
=-
FINN
FN
:
XPaAJ = O
.
FIN F(al 0
: = =
/(2)设 f (x)在区间[0,1]上连续,且 f (x) 0, x [0,1],证明
1 1
f 3(x)dx f 2(x)dx
0 0 .
1 1 # : * F' FIR
f 2(x)dx f (x)dx
0 0
e(f rax(-10'
fix 10
fix
[i] ax ax
LEDA (10 finax Y I . fix I
: = fix ax
fin I'
I !
fix
= ax ax
.
-
:1
例3.19 设I = lnn xdx，利用I 与I 的递推关系计算
n n n−1
0
I
n
.
*
f'llxax
l
! 10 Xn
LENA In x ax
: = = . -
↑ ↓
n)) (ax
ni2n
= - +
= -
nx( m- 1) Int = - nx( - (- ) ( - (n - z)[n - 3
= -
-
=- nx( - (n - 1) x( - (n - 2))x .. .. x ( 2)x(t)x
En !. S' lixax
En!
= =
U
#例3.20
下列积分发散的是 ( D )
O & 1 1
+ 1 +
2
(A) xe−x dx (B) x2 ln2 xdx (C) dx (D) 2 dx
0 0 e x(ln x )2 0 cos x sin x
B
/- Y ax JEYNEX (b 0)
& >
(A)
x
Mm
o
(B) = 0
.
I UT
I
(a) dx
.
X(c(x)
05 2
(D)
I
~
↳
Of
X+ AJ 3
. NosX
SiX
.例3.20
下列积分发散的是 ( D )
1 1
+ 1 +
2
(A) xe−x dx (B) x2 ln2 xdx (C) dx (D) 2 dx
0 0 e x(ln x )2 0 cos x sin x
E
2 - I I t = - X I
X- z ~ 1
=
loSX SMX losX cos(
. +) Sit
-
to
0
I I 45EX
=
N
.
E t1 e−x
+ +
例3.21 设 f (x)dx收敛,且 f (x) = − f (x)dx,求
0 1 + x2 1 + ex 0
+
f (x)dx.
1 i n2)
A * fixax
0 =
=
12 fin
: i A on
=
= Hirax-1A.
60 fiax
: x
A) (
E Hex 2 X + /
(ax Hexa
=> A = - - = + A . e +
~
to gX
A(
E ax E I
= - A + e + ( + et = - A + A . . He)dex
,
ex
+
In (n2A
= A
A + A = - + .
- .
Ite
ox
例3.22 设函数 f (x) = , x [0,1],定义函数列 f (x) = f (x),
1
1 + x
( ) ( )
f = f f (x) , f ( x) = f f (x) , 设S 是曲线 y = f (x),直线
2 1 n n−1 n n
x = 1, y = 0所围图形的面积.求极限limnS .
n
n→
fi * Y
ful fEfi]
fin
Ex
= = It X 1 *
= =
, fin = =
It &
It>X 1+2X
It
1+X
14
i
=
ful
#
X=
x
Su
>
=
=Co
x =
Su x
xax
Anatux(0)
= )1 C1-khtu)
- =n))
(1 mu 1
W U.S + =
-
=
S例3.23 计算抛物线 y = 3x − x2 与 x 轴围成的图形绕 x = −1旋转一周所成
的旋转体的体积.
( X
.
# 35 EX =: 2x +1) fix ax
: - :
D
I I
( ? 7
(3xx-xYax x
V
22 (+ 1) = 8 3
=
x= -
2
/rixy)
35 : V =
axay
=
(+ ) dx any
***
2x)ax1 2)?
(x+1) my (x + ) (3x x2)dx =
= = - x = acos3 t,
例3.24 (数一、数二)设星形线的方程为 (a 0)试求：它绕
y = asin3 t,
x 轴旋转而成的体积和表面积. X
/
: V *24ax
V 2 = 2
=
&
Glost
X
=
2/ lasma
Smit
/a sit absta
= 2
. =
a
=4xasmit.
22yds
vinylits
S 2Si = at
= =
a
=↑
例3.25 (仅数一、数二考)证明：曲线 y = sin x的一个周期的弧长等于
椭圆2x2 + y2 = 2的周长.
/
4/ity"ax
GERA 45
S =
: =
,
Si
-
1
8 X
↳
z
osdX
4 It
=
X lost
- =
zX
y + = 1 E
2x
+ = Y ESmt
=
(
42x
I
1 4) yiz)
= = at
+
.
