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: J2xetdx=Jerd(x)=et +Ce3v练习求dx.Vx解: 原式= 2Je3/F d/x=多Je3vd(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) [ f(ax+b)dx ==J f(ax+b)d(ax+b)万能凑幂法}J (x")dx"(2) [ f(x")x"-1 dx =(3) [ r(x)dx=(x)dx"(4) J f(sin x)cos xdx =J 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(tan x)sec’ xdx = J f(tanx)dtan x(7) [ f(e")e'dx = J f(e")de*(8) [f(nx)=dx=[ F(Inx)dlnx(9)] f(arcsin x)-dx = J f(arcsin x)darcsin x(10)] f(arctan x)dx = J f(arctan x)darctan xa一
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 = +
(11)] f(xe*)[e*(x +1)]dx = f(xe*)dxe*(12)] f(xIn x)(lnx + 1)dx = J f(x In x)dxIn x1xX(13)] f(dx=Jf()dV(1+x2)3/1+x1+x/1+x dx=[[df()2 =In|f(x) +C(11)]f(x)f(x)
(11) ( )[ ( 1)]d ( )d x x x x f xe e x x f xe xe + = (12) ( ln )(ln 1)d ( ln )d ln f x x x x f x x x x + = 2 2 3 2 2 1 (13) ( ) d ( )d 1 (1 ) 1 1 x x x f x f x x x x = + + + + ( ) ( ) (11) ln ( ) ( ) ( ) f x df x dx f x C f x f x = = +