How to prove ∫+∞−∞f(x)dx=∫+∞−∞f(x−1x)dx?\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x – \frac{1}{x}\right)dx?

If f(x) is a continuous function on (,+) and +f(x)dx exists. How can I prove that
+f(x)dx=+f(x1x)dx ?

Answer

We can write
f(xx1)dx=0f(xx1)dx+0f(xx1)dx=f(2sinhθ)eθdθ+f(2sinhθ)eθdθ=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=eθ in the second one.

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

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