Answer
$2 \int_{a}^{b} f(x)f'(x)dx = [f(b)]^2-[f(a)]^2$
Work Step by Step
$2 \int_{a}^{b} f(x)f'(x)dx$
Let $u = f(x)$
$\frac{du}{dx} = f'(x)$
$dx = \frac{du}{f'(x)}$
When $x = a$, then $u = f(a)$
When $x = b$, then $u = f(b)$
$2 \int_{f(a)}^{f(b)} uf'(x)\cdot \frac{du}{f'(x)}$
$=2 \int_{f(a)}^{f(b)} u~du$
$=2 (\frac{1}{2}u^2)~\Big\vert_{f(a)}^{f(b)}$
$=(u^2)~\Big\vert_{f(a)}^{f(b)}$
$=[f(b)]^2-[f(a)]^2$
