Vsin"x-sin'xdx.例3.计算解: : f(x)= /sin' x-sin'x =|cosx(sinx)2S* Vsin' x-sin' xdx = f" Icos xl(sin x)2 dxJe cos x(sin x) dx -J= cos x(sin x) dx (sinx) d(sin x) -[ (sin x) d(sin x)[(sin)][,(sin)]-
例3. 计算 3 5 0 sin sin d . x x x 解: 3 5 f x x x ( ) sin sin 3 cos sin x x 2 3 5 0 sin sin d x x x 3 2 0 cos sin d x x x 3 2 2 0 cos sin d x x x 3 2 2 cos sin d x x x 3 2 2 0 sin d(sin ) x x 3 2 2 sin d(sin ) x x 5 2 2 0 2 sin 5 x 2 5 2 2 sin 5 x 4 . 5
larcsinx练习．计算dx/x(1 -x)
练习. 计算 1 0 arcsin d . (1 ) x x x x
4x+2dx例4.计算J0/2x+1t2-解：令t=~2x+1,则x=dx=tdt, 且2当x=0时,t=1:x=4时,t=3原式=tdt=(c2+3)d223+3t)3
例4. 计算 解: 令 t 2x 1, 则 , d d , 2 1 2 x t t t x 当 x 0时, x 4时, t 3 . ∴ 原式 = t t t t d 3 2 1 2 1 2 (t 3) d t 2 1 3 1 2 3 ) 3 1 ( 2 1 3 t t 1 3 t 1; 且
偶倍奇零例5. 设 f(x)eC[-a,a],(1)若f(-x)= f(x), 则[~ f(x)dx=2f~f(x)dx(2)若f(-x)=-f(x), 则(α f(x)dx = 0证: [° f(x)dx = [° (x)dx + Jf(x)dx令x=-1= J°f(-t)dt + f°f(x)dxJ°l f(-x)+ f(x)]dx2f°f(x)dx, f(-x)= f(x)时0f(-x)=-f(x)时
例5. 证: (1) 若 a a a f x x f x x 0 则 ( ) d 2 ( ) d f x x aa ( ) d (2) 若 ( ) d 0 aa 则 f x x f x x a ( ) d 0 f x x a ( ) d 0 f t t a ( ) d 0 f x x a ( )d 0 f x f x x a[ ( ) ( )]d 0 f (x) f (x)时 f (x) f (x)时 偶倍奇零 令 x t