Definite integral of an absolute value function

[PFuser1232]

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.

 

[HallsofIvy]

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