Question

问题如图，在线等急！求大佬帮帮忙？

求解答过程

Answer 1

1/2∫(π/4,3π/4)sin^5θdθ
=-1/2∫(π/4,3π/4)(sin²θ)²dcosθ
=-1/2∫(π/4,3π/4)(1-cos²θ)²dcosθ
令t=cosθ
t1=cosπ/4=√2/2
t2=cos3π/4=-√2/2
上式=-1/2∫(√2/2,-√2/2)(1-t²)²dt
=1/2∫(-√2/2,√2/2)(1-2t²+t^4)dt
因为偶函数在对称区间的积分，等于2倍一半区间的积分。
所以，上式=∫(0,√2/2)(1-2t²+t^4)dt
=(t-2/3t³+1/5t^5)|(0,√2/2)
=√2/2-2/3(√2/2)³+1/5(√2/2)^5
=√2/2-√2/6+√2/40
=43√2/120

Answer 2

∫(sinx)^5dx =-∫(sinx)^4dcosx =-∫[1-(cosx)^2]^2dcosx =-∫[(cosx)^4-2(cosx)^2+1]dcosx =-(cosx)^5/5+2(cosx)^3/3-cosx+C C为积分常数
然后区间分成Π/4到Π/2，Π/2到Π/4，剩下的会了吧

Answer 3