帮忙算个定积分积分号0到二分之派 sinx^4cosx^4dx 答案是256分之3派.题目请看表述写下,

Question

帮忙算个定积分
积分号0到二分之派 sinx^4cosx^4dx 答案是256分之3派.题目请看表述写下,

Answer 1

∫(0,π/2)(sinx)^4*(cosx)^4dx
=(1/16)*∫(0,π/2)(sin2x)^4dx=(1/32)*∫(0,π)(sinx)^4dx
又∫(0,π)(sinx)^4dx
=-cosx*(sinx)^3|(0,π)+∫(0,π)3(sinx)^2*(cosx)^2dx=3*∫(0,π)[(sinx)^2-(sinx)^4]dx
∴∫(0,π)(sinx)^4dx=(3/4)*∫(0,π)(sinx)^2dx
原式=(1/32)*(3/4)*∫(0,π)(sinx)^2dx
=(3/128)*∫(1-cos2x)/2dx
=3π/256