Prove the following and use it to evaluate the integral:

Question

I want to prove that:$$\int_{-\infty}^\infty f(x)dx=\int_{-\infty}^\infty f\left(x-\frac1x\right)dx$$ And use the result of this proof to evaluate:$$\int_{-\infty}^\infty\frac{x^2}{x^4+1}dx$$

Answer 1

Here is how to apply it to $$\int_{-\infty}^\infty\frac{x^2}{x^4+1}dx = \int_{-\infty}^\infty\frac{1}{(x-\frac1x)^2+2}dx = \int_{-\infty}^\infty\frac{1}{x^2+2}dx=\frac\pi{\sqrt2} $$