# 电大高等数学基础形考作业参考答案

【高等数学基础】形考作业4答案

（一）单项选择题

⒈若)(x f 的一个原函数是x 1

，则=')(x f （D ）． A. x ln B. 21x - C. x 1 D. 32

x

⒉下列等式成立的是（D ）． A

)(d )(x f x x f ='? B. )()(d x f x f =? C. )(d )(d x f x x f =? D.

)(d )(d d

x f x x f x =?

⒊若x x f cos )(=，则

='?x x f d )(（B ）．

A. c x +sin

B. c x +cos

C. c x +-sin

D. c x +-cos ⒋

=?x x f x x

d )(d d 3

2（ B ）． A. )(3

x f B. )(3

2

x f x C. )(31x f D. )(31

3x f ⒌若

?+=c x F x x f )(d )(，则?

=x x f x

d )(1（B ）．

A. c x F +)(

B. c x F +)(2

C. c x F +)2(

D. c x F x

+)(1

⒍下列无穷限积分收敛的是( D ) A. ?

+∞

1

x dx B. dx e x

?+∞0 C. ?+∞1x

dx D. ?+∞12x dx

（二）填空题

⒈函数)(x f 的不定积分是dx x f ?)(．

⒉若函数)(x F 与)(x G 是同一函数的原函数，则)(x F 与)(x G 之间有关系式）c x G x F 常数()()(=-．

⒊=?x x d e d 2

2

x

e

⒋='?

x x d )(tan c x +tan ⒌若?+=c x x x f 3cos d )(，则=')(x f )3cos(9x -

?-=+3

3

5

d )21(sin x x 3 ⒎若无穷积分?∞+1d 1

x x p

⒈c x x d x x x x +-=-=??1sin )1(1cos d 1cos

2 ⒉??+==c e x d e x x

x x x 22d e

⒊??+==c x x d x x x x )ln(ln )(ln ln 1d ln 1

⒋c x x x xdx x x x x x ++-=+-=??2sin 4

1

2cos 212cos 212cos 21d 2sin

⒌??=+=++=+e 11e 121)ln 3(21)ln 3d()ln 3(d ln 3e x x x x x x

⒍414141212121d e 2102210210

2102+=--=+-=------??e e e dx e x e x x x x x x

41

221ln 2d ln 211

2e

1

+=-=??

e xdx x x x x x e e

⒏??+-=

--=+-=e e e e

x e dx x x x x x x 11

21e

1212

1

11ln 1d ln （四）证明题

⒈证明：若)(x f 在],[a a -上可积并为奇函数，则0d )(=?

-a

a

x x f ．

?

-----=-=--=-=a a

a

a

a

a

a

a

dt t f dt t f dt t f dx x f t

x )()()()(令

0)()()(=?-=????---a

a

a a

a a

dx x f dx x f dx x f 证毕

⒉证明：若)(x f 在],[a a -上可积并为偶函数，则??

=-a

a

a

x x f x x f 0

d )(2d )(．

???

+=--a

a

a

a

x x f x x f x x f 0

0d )(d )(d )(

???=--=-=-a

a

a

x f t f t f x x f t x 0

)(dt

)(dt )(d )(,是偶函数则令Θ

=+=+=--a

a

a

a a

a

a

x

x f x x f x x f x x f x x f x x f 0

0d )(2d )(d )(d )(d )(d )(

⒊证明：??

-+=-a

a

a

x x f x f x x f 0d )]()([d )(

+--=+=--a

a

a

a

a

a

x x f x x f x x f x x f x x f 0

d )(d )(d )(d )(d )(

=???

-+=+-a

a

a

x x f x f x x f x x f 0

d )]()([d )(d )( 证毕