求定积分sin^2x/(e^x+1)在(-π/2,π/2)的定积分

Question

求定积分
sin^2x/(e^x+1)在(-π/2,π/2)的定积分

Answer 1

令f(x)=sin^2x/(e^x+1)
f(-x)=e^x*sin^2x/(1+e^x)=e^x*f(x)
原式=∫(-π/2,0)f(x)dx+∫(0,π/2)f(x)dx
=-∫(π/2,0)f(-t)dt+∫(0,π/2)f(x)dx
=∫(0,π/2)f(-x)dx+∫(0,π/2)f(x)dx
=∫(0,π/2)f(x)*(e^x+1)dx
=∫(0,π/2)sin^2xdx
=1/2*∫(0,π/2)(1-cos2x)dx
=(x/2-1/4*sin2x)|(0,π/2)
=π/4