x练习dx2(2 +3x)u-2dx ==du令2+3x=u则x=33
练习 2d (2 3 ) x x + x 令2 3 = + x u 2 = 3 u x − 则 1 d d 3 x u =
例4 求[2xedx解: [2xetdx=erd(x)=et +Ce3v练习求dx.Vx解: 原式= 2Je3/F d/x=2Je3vd(3/x)32-3e3vr+C当被积函数一部分是另一部分(或其中主要部分）的导数时，直接凑微分
2 4 2 x xe dx 例 求 2 2 2 2 x x xe dx e xdx = 解 : 2 d x( ) 2 x = + e C 3 d . x e x x 解: 原式 = 3 2 dx e x 2 3 d(3 ) 3 x = e x 2 3 3 x = + e C 练习 求 当被积函数一部分是另一部分(或其中主要部分) 的导数时，直接凑微分
常用的几种配元形式：(1) J f(ax+b)dx ==J f(ax+b)d(ax+b)万能凑幂法}J (x")dx"(2) [ f(x")x"-I dx =(3) [ r(x)dx=(x)dx(4) J f(sin x)cos xdx =[ f(sin x)dsin x(5) J f(cos x)sin xdx = -J f(cosx)dcosx
常用的几种配元形式: (1) ( )d f ax b x + = 1 f ax b a x b ( )d( ) a + + 1 (2) ( ) d n n f x x x − = 1 ( )d n n f x x n 1 (3) ( ) d n f x x x = 1 1 ( ) d n n n f x x n x 万 能 凑 幂 法 (4) (sin )cos d f x x x = f x x (sin )dsin (5) (cos )sin d f x x x = − f x x (cos )dcos
(6) J f(tanx)sec2 xdx = f(tanx)dtanxJ f(x)sec x tan xdx =f f(x)d(sec x)(7) [ f(e*)e*dx = [ f(e")der(8) [ f(ln x) =dx = [ f(ln x)dln x(9)] f(arcsin x)-dx = J f(arcsin x)darcsin x(10)J f(arctan x)redx = J f(arctan x)d arctan x+x
2 (6) (tan )sec d f x x x = f x x (tan )dtan (7) ( ) d x x f e e x = ( )d x x f e e 1 (8) (ln ) d f x x x = f x x (ln )dln 2 1 (9) (arcsin ) d (arcsin )darcsin 1 f x x f x x x = − 2 1 (10) (arctan ) d (arctan )darctan 1 f x x f x x x = + f x x xd x f x d x ( )sec tan ( ) (sec ) =
(11) Jf(x)e*(1+x)dx= J f(x)d(xe")(12) [ f(x)(ln x+1)d x=J f(x)d(x ln x)(13) [ f(x) 一dx =2J f(x)d(Vx)Xx(14)] f(x)dx = [ f(x)dVx* ±a?Vx'±ax(15) f(x)dx =±[ f(x)d/a' ±x±x?Va'(16)J f(x)sin 2xdx =J f(x)dsin’ x=-J f(x)dcos* x
(11) ( ) (1 )d ( ) ( ) x x f x e x x f x d xe + = (12) ( )(ln 1)d ( )d( ln ) f x x x f x x x + = 1 (13) ( ) d 2 ( )d( ) f x x f x x x = 2 2 2 2 (14) ( ) d ( )d x f x x f x x a x a = 2 2 (16) ( )sin 2 d ( )dsin ( )dcos f x x x f x x f x x = = − 2 2 2 2 (15) ( ) d ( )d x f x x f x a x a x =