If f(x) is a continuous function on (−∞,+∞) and ∫+∞−∞f(x)dx exists. How can I prove that

∫+∞−∞f(x)dx=∫+∞−∞f(x−1x)dx ?

**Answer**

We can write

∫∞−∞f(x−x−1)dx=∫∞0f(x−x−1)dx+∫0−∞f(x−x−1)dx=∫∞−∞f(2sinhθ)eθdθ+∫∞−∞f(2sinhθ)e−θdθ=∫∞−∞f(2sinhθ)2coshθdθ=∫∞−∞f(x)dx.

To pass from the first to the second line, we make the change of variables x=eθ in the first integral and x=−e−θ in the second one.

**Attribution***Source : Link , Question Author : Hung Nguyen , Answer Author : Larry*