不定积分的概念与性质

§4.1 不定积分的概念与性质

阶段练习题(A)

一、选择题

1.下列等式中正确的是( ). (A)d(

()d )()f x x f x =⎰; (B)d

[d ()]()d d f x f x x x =⎰; (C)d ()()f x f x =⎰; (D)()d ()f x x f x C '=+⎰.

2.设函数

(),x

f x a =()(0,1)ln x

a g x a a a

=>≠,则( ).

(A)

()g x 是()f x 的不定积分; (B)()g x 是()f x 的导数;

(C)

()f x 是()g x 的原函数; (D)()g x 是()f x 的原函数.

3.

()f x 的一个原函数为ln x ,则()f x '=( ).

(A)

1x

; (B)

ln x x x C -+; (C)2

1x -

; (D)x

e .

二、填空题

1. (

5

sin d )x

x x '=⎰ . 2.d(arctan )x =⎰ .

3.

()f x 的原函数是2ln ,x 则3()d x f x x '=⎰ .

4.设

21(),cos f x x

=

则()d f x x '=⎰ ,d

()d d f x x x =⎰ ,()d f x x =⎰ . 5.设

()d ,x

x f x x xe

e C =-+⎰则()d

f x x '=⎰ . 6.设()f x 的一个原函数为1

,x

则()f x '= .

7.过点(0,1)且在横坐标为

x 的点处的切线斜率为3x 的曲线方程为 .

8.设

22(cos )sin ,f x x '=且(0)0,f =则()f x = . 9.21

(

1)dcos cos x x

-=⎰ .

三、解答题

求下列不定积分:

1.

x ; 2.2

1

(1x x -

⎰; 3.21d 1x x e x e -+⎰; 4.221d sin cos x x x ⎰;

5.

327d

3x x x --⎰; 6.4x ;

7.

221d (1)x x x +⎰; 8.2sin d 2

x x ⎰; 9.

2

cot d x x ⎰; 10.d 1cos 2x

x

-⎰

;

11.

22d 1x x x +⎰; 12.2d x x

e x ⎰.

§4.2 换元积分法

阶段练习题(A)

一、选择题

1.

d t =( ).

(A)C -; (B)C ; (C)C -; (D)C .

2.

d x x x

e e -=+⎰( ).

(A)arctan x

e C +; (B)arctan x e C -+;(C)arccot x e C +; (D)arccot x e C -+.

3.已知

()x f x e -=,则(ln )

d f x x x

'=⎰

( ).

(A)1C x

-

+; (B)

1

C x

+; (C)ln x C -+; (D)ln x C +. 4.

21ln d (ln )x

x x x +=⎰( ).

(A)

1ln C x x +; (B) 1ln C x +; (C)1ln C x x -+; (D)1

ln C x

-+.

5.

10

2tan

sec d x x x =⎰( ).

(A)111tan 11x C -

+; (B)111tan 11x C +; (C)101tan 10x C -+; (D)101

tan 10

x C +.

二、填空题

1.d x = d (83)x -.

2.d x x = 2d ()x .

3.4d x

e

x = 4d ()x e . 4.3cos d 2x x = 3

d (sin )2

x .

5.

1d x x = d (10ln )x . 6.2

1

d 116x x =+ d (arctan 4)x .

x = d (2arcsin )x -x = d .

9.

2

1

d 1x x -=+ d (2arccot )x -x = d . 三、解答题

求下列不定积分:

1.

1d 12x x

+⎰. 2.3sin d cos x

x x ⎰.

3.

x

⎰. 4.

3

cot

csc d x x x ⎰.

5.

d ln ln ln x

x x x ⎰. 6.

23d 94x x

x x x ⋅-⎰.