# Definite integral of an absolute value function

1. Jun 25, 2015

Can we integrate:
$$\int_a^b |x| dx$$
using an antiderivative of $|x|$, namely $\frac{1}{2} x |x|$, instead of splitting up the integration interval?
I know this is not particularly useful for integrals such as:
$$\int_{-5}^5 |t^3 - 8| dt$$
However, for absolute value functions with linear arguments, this method (if valid) would be much more efficient.

2. Jun 25, 2015

### HallsofIvy

Staff Emeritus
Yes, of course. If F(x) is an anti-derivative of f(x) then $\int_a^b f(x) dx= F(b)- F(a)$. That is true for f(x)= |x| and F(x)= (1/2)x|x|.