Indefinite integrals with absolute values

Question

Which is the right way to solve indefinite integrals which contain absolute values? For example if I have $\int |2x+3| e^x dx$
Can I consider the sign function and integrate separetly? I mean doing: $ Sign(2x+3) \int (2x+3)e^x dx$

Or maybe I should use the definition of absolute value and divide the two possibilities
$\int (2x+3)e^x dx$ if $ (2x+3)>0$ and $\int (-2x-3)e^x dx$ if $ (2x+3)<0$

But I think that's more suitable for definite rather than indefinite integrals

How can I solve this type of integrals? Thanks a lot in advice

Answer 1

HINT: i would consider the cases $$2x+3\geq 0$$ or $$2x+3<0$$

Answer 2

$\int|2x+3|e^x~dx$

$=\text{sgn}(2x+3)\int(2x+3)e^x~dx$

$=\text{sgn}(2x+3)\int_{-\frac{3}{2}}^x(2x+3)e^x~dx+C$

$=\text{sgn}(2x+3)\int_{-\frac{3}{2}}^x(2x+3)~d(e^x)+C$

$=\text{sgn}(2x+3)[(2x+3)e^x]_{-\frac{3}{2}}^x-\text{sgn}(2x+3)\int_{-\frac{3}{2}}^xe^x~d(2x+3)+C$

$=\text{sgn}(2x+3)(2x+3)e^x-2~\text{sgn}(2x+3)\int_{-\frac{3}{2}}^xe^x~dx+C$

$=|2x+3|e^x-2~\text{sgn}(2x+3)[e^x]_{-\frac{3}{2}}^x+C$

$=|2x+3|e^x-2~\text{sgn}(2x+3)\left(e^x-e^{-\frac{3}{2}}\right)+C$