可以取f(x)=1/5 sinx的五次方,则f'(x)=sinx的4次方,这样就可以积分啦!

∫ (sinx)^6 dx=∫ [(sinx)^2]^3 dx=(1/8)∫ [ 1- cos2x ]^3 dx= (1/8)∫ [ 1- 3cos2x + 3(cos2x)^2 - (cos2x)^3 ]dx=(1/8)[ x - (3/2)sin2x] +(3/8)∫ (cos2x)^2 dx-(1/8)∫ (cos2x)^3 dx =(1/8)[ x - (.

in=∫(0,π/2)[cos(x)]^ndx=∫(0,π/2)[sin(x)]^ndx =(n-1)/n*(n-3)/(n-2)*…*4/5*2/3,n为奇数; =(n-1)/n*(n-3)/(n-2)*…*3/4*1/2*π/2,n为偶数

首先有1-2sin²x=cos2x2cos²x-1=cos2x cosxcosy=1/2cos(x+y)cos(x-y) sin⁶x=1/8(1-cos2x)³(1-cos2x)³=1-3cos2x+3cos²2x-cos³2x=1-3cos2x+3/2(cos4x+1)-1/2(cos4x+1)cos2x=5/2-7/2cos2x+3/2cos4x-1/2cos2xcos4x=5/2-15/4cos2x+3/2cos4x-1/4cos6x ∴sin⁶x=1/32[10-15cos2x+6cos4x-cos6x] ∴原式=1/32[10x-15/2sin2x+3/2sin4x-1/6sin6x]+C

∫cos2xcos3xdx=1/2∫[cosx+cos5x]dx=1/2[sinx+1/5sin5x]+c=1/2sinx+1/10sin5x+c

I1=∫(sinx)^6 dx=-∫(sinx)^5 dcosx = -cosx (sinx)^5 + ∫cosxd(sinx)^5 =-cosx (sinx)^5 + 5∫cosx(sinx)^4 cosxdx =-cosx (sinx)^5 + 5∫(sinx)^4 - (sinx)^6dx6I1 =-cosx (sinx)^5 + 5∫(.

令x=pi/2+t,则t=x-pi/2,dt=dx sin(x)dx=sin(t+pi/2)^6=cost^6dt,然后使用积分公式

正弦积分是由积分定义的一种特殊函数[1] .英文翻译为sine integral,常用语计算正弦积分函数值,在性质上常与余弦积分作比较.正弦积分函数的定义为:,本函数采用勒让德-高斯求积公式计算该积分.

cos^6x=(cos²x)³=(1+cos2x)³/8=1/8(1+3cos2x+3cos²2x+cos³2x) cos²2x=1/2(1+cos4x) ∫cos^6xdx=1/8∫(1+3cos2x+3cos²2x+cos³2x)dx=1/8x+3/16sin2x+3/16∫(1+cos4x)dx+1/16∫cos²2xdsin2x=1/8x+3/16sin2x+3/16x+3/64sin4x+1/16∫(1-sin²2x)dsin2x=5x/16+15sin2x/16+1/16x-1/48sin³2x+c=3x/8+(15sin2x)/16-(sin³2x)/48+c

∫(sinx/2)^2dx =∫(1-cosx)/2 dx =(1/2) ∫1 dx- (1/2)∫cosx dx =x/2-(1/2)sinx