例6求∫ tan xdx 解∫andc=∫ sInx dr=-(cos r)dx cos x cos x dcos=--dr cos r =-In u/ +C=-Incos x+C 例7求 va=x 解∫ d va=x du arcsin+G arcsin -+c
tan . 求 xdx tan xdx = x x x d cos sin = − x x cos cos 1 d = − ln cos x + C. 例6 解 ( ) − = x x x cos cos d = − u u d 1 = − ln u + C . 1 2 2 − dx a x 求 − x a x d 2 2 1 x a x a d − = 2 1 1 1 arcsin C. a x = + a x a x d − = 2 1 1 例7 解 u u d − = 2 1 1 = arcsin u +C
例8求 解 (x-a)(x+a) dx Inlx-a-Inlx+a+C 2ax-a x+a d 例9求 x(1+2Inx 解∫ d x din x d2 Inx x+2n)-」(+2m)=2(+2m I rd(1+2In x 2(1+2lnx)-2 In/1+2Inx+c
. 1 2 2 − x x a 求 d − x x a d 2 2 1 − + = x x a x a d ( )( ) 1 + − − = x a x a x a d 1 1 21 (ln ln ) . 21 x a x a C a = − − + + 例 8 解 ( ) . 1 2ln x + x dx 求 ( ) x + x x 1 2ln d ( ) + = x x 1 2ln d ln ( ) + = xx 1 2ln 2ln 21 d ( ) ++ = xx 1 2ln ( 1 2ln ) 21 d ln 1 2ln . 21 = + x + C 例 9 解
3 例10求∫dx 3√x 2 解∫9c2dNx= e3xd3√x = +c 3 例11求im3xdr #f sin'xdx= sin xsin xdx 2 sin rd cosx - cOs cosx cos式 =-coSX十 +c
. 3 x x x d e 求 x x x d e 3 x x e d 3 2 = . 3 2 3 C x = e + = x x 3 3 2 3 e d sin . 3 求 xdx xdx 3 sin = sin xsin xdx 2 = − sin x cos x 2 d ( ) = − 1− cos x cos x 2 d . 3 cos cos 3 C x = − x + + 例10 解 例11 解
例12求∫ sin xcos xdx 解「im2 xcos xdx=sm2 xcos'xcos xdx sin2x1-sin x ∫sm小-2 sinx+sinx)smx sin2x-2sin4x +sinx Hsin sin'x--sin'x+-sin x+C 5
sin cos . 2 5 求 x xdx x xdx 2 5 sin cos = sin xcos xcos xdx 2 4 ( ) = sin x 1− sin x sin x 2 2 2 d ( ) = sin x 1− 2sin x + sin x sin x 2 2 4 d ( ) = sin x − 2sin x + sin x sin x 2 4 6 d sin . 7 1 sin 5 2 sin 3 1 3 5 7 = x − x + x + C 例12 解
例13求 cos xar 解Jc3xdr=+ cos 2 2 dx+ cos 2xdx dx+= cos 2xd2x x+sin 2x +C
cos . 2 求 xdx xdx 2 cos + = x x d 2 1 cos2 ( ) = dx + cos 2xdx 2 1 = + x cos 2x 2x 2 1 2 1 d d sin2 . 2 1 2 1 x x + C = + 例13 解