dr例3. 求(a >0).dxdxdx解：AX= arcsin=+ Cadu想到arcsinu+C1[ f[o(x)0(x)dx = [ f(0(x)dp(x)（直接配元）福目录上页下页返回结束机动
例3. 求 = − 2 1 d u u 想到 arcsin u + C 解: − 2 1 ( ) d a x a x = f ( (x))d (x) (直接配元) f [ (x)] (x)dx − = 2 1 ( ) d ( ) a x a x
例4. 求[ tan xdx.dcosxsinx解：tan xdx =dx =cosxcosx=-In cos x|+ C类似cosxdxdsinxcot xdxsinxsinx= In sinx+C上页目录下页返回结束机动
例4. 求 解: x x x d cos sin = − x x cos dcos x x x sin cos d = x x sin d sin 类似
dx例5.求解：1(x+a)-(x-a) l(2a (x-a)(x+a)2ax-ax+a[-[ a原式x+ad(x-α) -{d(x+a)[x+ax-ax-a-[1n x-a|-In x+al +C=二nC2a2.0x+a目录上页下页返回结束机动
C x a x a a + + − = ln 2 1 例5. 求 解: 2 2 1 x − a (x − a)(x + a) ( x + a) − ( x − a) 2a 1 = ) 1 1 ( 2 1 a x a x + a − − = ∴ 原式 = 2a 1 + − − x a x x a dx d = 2a 1 − − x a d(x a) 2a 1 = ln x − a − ln x + a + C + + − x a d(x a)
常用的几种配元形式(1) [ f(ax +b)dx =f(ax+b) d(ax +b)万能凑幂法(2) J f(x")xn-I dx :[f(x") dxn(3) J f(x")dx=(r")dxn(4) J f(sin x)cos xdx =J f(sinx)dsinx(5) J f(cos x)sin xdx = - J f(cos x) dcosx目录上页下页返回结束机动
常用的几种配元形式: + = (1) f (ax b)dx d(ax + b) a 1 = − f x x x n n (2) ( ) d 1 n dx n 1 = x x f x n d 1 (3) ( ) n dx n 1 n x 1 万 能 凑 幂 法 = (4) f (sin x)cos xdx dsin x = (5) f (cos x)sin xdx − dcos x
(6) f f(tan x)sec xdx = [ f(tanx) dtan x(7) [ f(e*)e*dx=J f(e") de(8) J f(Inx)-dx = f(nx) dlnxXdx例6. 求x(1+2lnx)dlnxd(1 + 2ln x)解：原式=J1+2lnx1+2lnx=In|1+2 lnx|+C目录上页下页返回结束机动
= (6) f (tan x)sec xdx 2 dtan x = f e e x x x (7) ( ) d x de = x x f x d 1 (8) (ln ) dln x 例6. 求 1+ 2ln x dln x 解: 原式 = + = 2 1 2 ln x 1 d(1 + 2 ln x)