文档内容
第三章
一元函数积分学x x
例3.1 设 f (sin2 x) = , x 0, ，求 f (x)dx.
sin x 2 1 − x
At
smix E arcsi
Smx X
= = =
, ,
arcsm aresmox
fl fix
= =
=
E #
comit
* -fixarsmac
fix
dX
: -
= -x)
*
↑ ↓
2) .
'dx
arsmix
+ x
= 21-x - ·
-
2
X
1 -
arsmOx 2x
-21-X + + C
= .↑
dx
例3.2 计算 .
sin 2x + 2sin x
(25mx(0x
1/2 smxos dX
=
+ )
zsmx
(acosx
sax
)
= 1)aX = -
Smix. (osx
2 + 2 (l-co5X) (10SX + 1)
drost dt
If E = losX * )
=
Kosxtlosx-1) (t + ll (t -1)
I b
a
C
+ t
=
P(t
(t + - 1) T -1 [t) E+ /12
I
+ -( +c I
a = =
= = - z
( = t-1t
=-(ty c yy) = -
b (
= = -
= + +- -
e - +]
/Te
T at
· -
=
(n(t 1) (n(t + ) c ++ + C
- - +
=
-
I
cosX-1
10sX C
- In + +
=
cosxti + /1 ( )
例3.3 设积分P = tan2 x + ex cos x − e−x cos x dx,
−1
( ) x
1
N = 4 sin4 x + ln x + x2 + 1 dx, M = + x2 dx,则正确的结
− −1 1 + x6
mem
4
F
论是(
D
).
(A)P N M (B) N P M (C) N M P (D)P M N
(_, Emix Lex eX) 10sx] % Tmixdx
P + ex = 2
= - .
2 Smix
2 x ax
ax M
N =
=
2
Smix Six <*<
. AF
E XeCo < tex
,
N
S M > P
[o , ] C [0 1] [0 1]
. .1
例3.4 设 f (x)在[0,1]上连续,且 f (x) 0, M = f (x)dx,
0
N = 2 f (sin x)dx, P = 4 f (tan x)dx,则( B ).
0 0
(A)M N P (B)P M N (C) N P M (D)M P N
So fin
dx
m =
Esft
fismax =
=
N
= fimx)axtuxfaf
p
/
McN
: P =
例3.5 设 f (x)在区间 0, 上单调、可导且满足
4
cost − sin t
f (x) x
f −1(t)dt = t dt ，其中 f −1 是
0 0 sint + cost
f 的反函数，求 f (x).
ik E
En I
losX-SmX
fix
fl fN] X
=> · = .
SiX + losX
losX-smx
fix
=> X . = X .
SmX + 10SX
losX SiX
-
=fix
E #F
X + 0 =
,
SmX
+loSX/los-smax )dBmx o
fix
In(SmX
: = = C
=
+osX +
SmX+ 10SX
(f)f
/X
73
* =0 =
((at = 0
f [0 ]
X
,
, ]
f TETEX [0
-
fix
: = o
fi . ] F = L
To
i
fol
Ifix
· = = 0 = C = 0 = f InIsmxty
=
例3.6 F(x)是 f (x)的一个原函数,F = 0,当 x 时,F(x) 0,且
4 4 2
ln(tan x)
F(x) f (x) = ,则 f (x) =________.
sin xcos x
F(x fix)
#BEER =
,
I
(InCtm [In(tmx)] " seix =Y
) FN) Fix ax a dX = ↑
: =
co5X
Tex smX
/FINdFi /Intanx
d (I mx)
+
=> =
(Intaxi
Fix
I & C
=> +
=
F(2)
· = 0 i C = 0mx)"
Fix Du
+
=
:
Intax
FM
=
&
fix
c
=
SmX-10SX
例3.7 如图，连续函数 y = f (x)在区间 −3,−2 , 2,3 上的图形分别是直
径为 1 的上、下半圆周，在区间 −2,0 , 0,2 上的图形分别是直径为 2
x
的上、下半圆周.设F(x) = f (t)dt,则下列结论正确的是( C )
0
3 5
(A)F(3)= − F(−2) (B)F(3)= F(2)
4 4
3 5 4S
S
(C)F(−3) = F(2) (D)F(−3) = − F(−2)
O 3
-3 -2 4-13 1 2
S
4 4
1 fitdt 3 S
F(3) = = F(1 F(2)
= =
F(2)
/ .fitIdt
=
12 fit
F(2) at = 43
=
[fiat
-
=
to fit
fi Fin FE31 F(3)
- = =
F( 2) F(z)
=
例3.8 设函数 y = f (x)在区间 −1,3 上的图形如图所示，则函数
x
F(x) = f (t)dt 的图形为 ( ).
D
0
X(A) X (B)
# F(d
= 0 = (c) X
# F(xYY(X
= (B) X
X
(C) (D) # [ -1 0] 177
.
F(x f(x)
10
= =
F(x)T例3.9 设 f (x)在[0,]上具有二阶连续导数, f () = 2,且
f (x) + f (x) sin xdx = 5, 则 f (0) =___3_____.
0
I fin
1f'm.
↓"I fix fix] smxax suxax + smxdx
= -
+ -
↓ ↑ ↓
+ fixcoxax
fi.cosX? fix * -* fixcox
Six
+ ax
·
=
=
[ - f(x) f(0)] f(x f(0) f(0)
= - - = + = 2 + = 5
fiol
3
in =
例3.10 计算定积分I = 4 ln(1 + tan x)dx.
tmd-taB
tm/d-Bl
0
=
* BEFE It tod tmB
] IB. E * T .
[ I :
.
2
-
=
= I- x ( · tmt
m -t))Edt) InCHtmE1
: In (H at
+
.
F
-
X = 4 t E It Tm tant
(lat
=
dt
Ittrut
In2-In
[In2-In Chtet)] at (H ++mx)ax
=
=
W
In2-I
=
In
(n2 :T
: 21
= . =x+2
例3.11 设F(x) = esint sintdt ，则F(x)( A ).
x
(A)为正常数 (B)为负常数 (C)恒为零 (D)不为常数
35-i
esh? IT HE In
Sme
-
= = Teshtsat-e-sht
F espesmeat
suit) at
-
D
Leshtesht
"
= see at >o
10
=
30x+2
例3.11 设F(x) = esint sintdt ，则F(x)( ).
A
x
(A)为正常数 (B)为负常数 (C)恒为零 (D)不为常数
eShtcost1st
1eSht sh cost
FN
35
=: = + de = -
↓ ↑
(stat
=
>
7070x+2
例3.11 设F(x) = esint sintdt ，则F(x)( A ).
x
(A)为正常数 (B)为负常数 (C)恒为零 (D)不为常数
/esmt
FI at
35= sat
: =
et1"/11
# ~
sut
[0in ] / Sito
,
a
FTP eshte 0 7 -211/
[2 2) 17 sunt =0 ,
. ,
,1 x
st
例3.12 设函数 f (x)连续,I = f t + dx,其中s 0,t 0,则().
s 0 s
(A)依赖于s,t 和 x . (B)依赖于 s 和 t .
(C)依赖于t ,不依赖于 s. (D)依赖于 t 和 x
~
,不依赖于 s.
+
2 f(t + )axn=
fusan =1
fill
ac
I
du dX
= 52 3
2
例3.13 设 f (x) = e− y dy,计算I = f (x)dx.
x−1 1
33 7 /1 fax Xfm ? 1X- fix ax
- = = . = -
?
↑ ↓
11) ax
-(3x) (
fi e-
fix)
-
3
= -
-
eTay
( : (xe - (
= o - + ax
+
(2x ( 1 = x+ (2(4 - Tay
=-
ax + 1) E
.
Yay
em et
-1: 1024t 10
1 m
: +
= =
easy
=M
1: =(1 4)
e
=- = - = -2 3
2
例3.13 设 f (x) = e− y dy,计算I = f (x)dx.
x−1 1
(leay)ax 1 : ax(_ -May
1 1?
? faux =
35 =: 1 = =
-eaxay 1 ? ay) Ex
↑
= (3 2)
· .
D5Y X
=
=
I ey
:
=
i's
·
e-4)
(1
= -
