Question

求定积分高数

Answer 1

原式=∫(-π/2,π/2)(sinx)^2/[1+e^(-x)]dx
=∫(-π/2,0)(sinx)^2/[1+e^(-x)]dx+∫(0,π/2)(sinx)^2/[1+e^(-x)]dx
=-∫(π/2,0)(sinx)^2/(1+e^x)dx+∫(0,π/2)(sinx)^2/[1+e^(-x)]dx (第一个积分用-x代换x得)
=∫(0,π/2)(sinx)^2/(1+e^x)dx+∫(0,π/2)e^x(sinx)^2/(1+e^x)dx (第二个积分分子分母同乘e^x得)
=∫(0,π/2)(1+e^x)(sinx)^2/(1+e^x)dx
=∫(0,π/2)(sinx)^2dx
=1/2∫(0,π/2)[1-cos(2x)]dx
=1/2[x-1/2sin(2x)]|(0,π/2)
=1/2(π/2-0)
=π/4
结果自己在计算一下。

Answer 2

追问

牛逼

我也是这样想的可是总是化不出来，还是太却经验了。