Question

高数，定积分这几部怎么推导的？

划线地方，蓝色线，蓝色线，蓝色线

Answer 1

arctanx +arctan(1/x) = π/2
u=-x
du =-dx
I
=∫(-π/4->π/4) 5sinx. [e^x/(1+e^(2x)] dx

=∫(π/4->-π/4) -5sinu. [e^(-u)/(1+e^(-2u)] ( -du)
=∫(-π/4->π/4) -5sinx. [e^(-x)/(1+e^(-2x)] dx
=∫(-π/4->π/4) -5sinx. [e^(x)/(1+e^(2x)] dx
2I=∫(-π/4->π/4) 5sinx. [e^x/(1+e^(2x)] dx+∫(-π/4->π/4) -5sinx. [e^(x)/(1+e^(2x)] dx
=0
I=0

Answer 2

如下图所示，换元法，变换被积函数，然后用三角函数和指数函数的基本性质来做: