Integration of $f(-x)$

Question

What happens to the sign and bounds of the integral when you want to change $f(-x)$ to $f(x)$? Is the following correct? How does one convert from $xf(-x)$ to $xf(x)$? $$\int_{0}^{\infty} xf(-x)\ dx = -\int_{\infty}^{0} xf(x)\ dx\; ?$$

Answer 1

Let consider the change of variable $y=-x \implies dx=-dy$ then

$$\int_{0}^{\infty} f(-x)\ dx=-\int_{0}^{-\infty} f(y)\ dy=\int_{-\infty}^0 f(y)\ dy$$

Answer 2

If $f(x)$ is an even function then no changes will take place in the upper and lower bounds but if $f(x)$ is an odd function then you can trap the minus sign and interchange the upper and lower bounds to get the answer.