《高等数学》课程教学资源（知识与题解PPT）5.1.2 微分的逆运算问题——不定积分（基本积分公式）

s12基本积分公式 F"(x)=f(x) f(x)dx= F(x)+C +1 例如:(x,)=x(≠-1) + +1 →|xar A+1+C

§1.2 基本积分公式 f x dx = F x +C  ( ) ( ) F (x)=f(x) 例如:  = + + ) 1 ( 1   x x  C x x dx + +  = +  1 1    (   −1)

基本积分表 (1)0x=C ( 2) dx=x+C (3)x4dx= -b4 H+1C(≠-1) (4)1x=ln|x|+C(x≠0) (5)ex=e2+C

基本积分表 dx = x +C  (2) C x x dx + + = +  1 (3) 1    dx = C  (1) 0 (   −1) dx x C x = +  ln | | 1 (4) ( x  0) e dx e C x x = +  (5)

(6)ax=+C(a>0,a≠1) (7I cos xdx=sinx+C (8) sin xd=-cosx+C ) sec xdx=」∫3d=nx+C (10) csc2xdx= 1dx =-cotx+c sin x

C a a a dx x x = +  ln (6) (a >0, a  1) xdx = x +C  (7) cos sin xdx = − x +C  (8) sin cos dx x C x xdx = = +   tan cos 1 (9) sec 2 2 dx x C x xdx = = − +   cot sin 1 (10) csc 2 2

(Dlsecxtan xdx=secx+C (12 )cscxcot xdx=-cscx+C (13 dx arcsin x+c arccos+C 14)「,12k= arctan x+C 1+x -arccot+C

x xdx = x +C  (11) sec tan sec x xdx = − x +C  (12) csc cot csc dx x C x = + −  arcsin 1 1 (13) 2 = −arccosx+C  dx x C x = + +  arctan 1 1 (14) 2 = −arccotx+C 

例1求积分x2√xdx 解:」x rax xide A+l 根据积分公式「xdx=x,;+ + x2√xdx= x52 +C=2x2+C

例1 求积分  x xdx 2 解:  x xdx 2  = x dx 2 5 C x x dx + + = +  1 1    根据积分公式  x xdx 2 C x + + = + 1 2 5 1 2 5 = x 2 +C 7 7 2