例6若(x)在[0,1上连续,证明 (1)2 f(sinx)dx=2 f(cosx)dx (2)[ xf(sin x)dx3 y(Sinxydx 证明(2)令x=x因为 So xf(sinx)dx=-(T-D)/[sin(T-t)lt l(/Tsin(T-t)kt=(7-o)f(sint)dt rS f(sint)dt-ltf(sindt r Jo f(sinxydx-Jo f(sinx)dx 所以y( (sin x )dx=xf(smxk 上页返回 页结束铃首页 上页 返回 下页 结束 铃 (2)令x=−t 因为 例6 若f(x)在[0, 1]上连续, 证明 (2)   =    0 0 (sin ) 2 xf (sin x)dx f x dx  (1)   = 2 0 2 0 (sin ) (cos )   f x dx f x dx ; 证明   =− − − 0 0 (sin ) ( ) [sin( )]   xf x dx  t f  t dt   = − − = −      0 0 ( t)f[sin( t)]dt ( t)f (sint)dt   = −    0 0 f (sint)dt tf (sint)dt   = −    0 0 f (sin x)dx xf (sin x)dx 所以   =    0 0 (sin ) 2 xf (sin x)dx f x dx    =− − − 0 0 (sin ) ( ) [sin( )]   xf x dx  t f  t dt   = − − = −      0 0 ( t)f[sin( t)]dt ( t)f (sint)dt 下页