$\int _{-\infty}^{\infty}f(x)dx=\int _{\infty}^{-\infty}f(-x)d(-x)$

Question

Let $f(x)$ be a continuous function on $\mathbb{R}$, is $$\int _{-\infty}^{\infty}f(x)dx=\int _{\infty}^{-\infty}f(-x)d(-x)$$ true? If it's true, please prove it.

Answer 1

If the integral $\int_{-\infty}^{\infty} f(x) \mathrm{dx}$ exists, then by substitution $t = -x$, you get immediately your equality.

If the integral doesn't exist, then your equality has no sense.