Absolute value integral in terms of $x$ as a function of $a$.

Question

Compute $F(a)=\int_{0}^{1}\left|x^2+a\right|dx$

We know that:

$$\left|x^2+a\right|=\left\{\begin{array}{l}x^2+a,a\geq x^2\\-x^2-a,\;a<x^2\end{array}\right.$$

But, when this is under the integral sign do I have to integrate the inequalities too?

Answer 1

Do it in separate cases according to the value of $a$.

1. If $a\ge0$, then $|x^2+a|=x^2+a$.
2. If $a\le-1$, then $|x^2+a|=-a-x^2=|a|-x^2$.
3. If $-1<a<0$, then divide the interval of integration in two parts: $$\int_0^1=\int_0^{\sqrt{|a|}}+\int_{\sqrt{|a|}}^1.$$