How to prove the definite integral for even functions

Question

I'm having trouble with the following proof:

If f is an even function on [−a, a] then $\int_{-a}^{a} f(x)dx$ = $ 2 \int_{a}^{0}f(x)dx$

$\int_{-a}^{a}f(x)dx$

= $\int_{-a}^{0}f(x)dx$ + $\int_{0}^{a}f(x)dx$

= $\int_{-a}^{0}f(-x)dx$ + $\int_{0}^{a}f(x)dx$

Let u = -x du = -dx dx = -du

= $-\int_{-a}^{0}f(u)du$ + $\int_{0}^{a}f(x)dx$

= $\int_{0}^{-a}f(u)du$ + $\int_{0}^{a}f(x)dx$

I get stuck after this step. Can anyone please help me out?

Answer 1

\begin{align*} \int_{-a}^{0}f(-x)dx=-\int_{a}^{0}f(u)du=\int_{0}^{a}f(u)du=\int_{0}^{a}f(x)dx. \end{align*}