Integral of the product of a function and its derivative.

Question

Say I have the indefinite integral $\int g(x)g'(x)~dx$, how would I go about solving this?

Answer 1

Notice that $2g'(x) g(x) = [g(x)^2]'$

So

$$\int g'(x) g(x) dx = \int [\frac{g(x)^2}{2}]' dx = \frac{g(x)^2}{2}$$

As the primitive of the derivative of a function is this function

Answer 2

If we write $u=g(x)$ then $du=g'(x)dx$ and so

$$\int g(x)g'(x)~dx=\int udu=\frac{u^2}{2}+c=\frac{[g(x)]^2}{2}+c$$