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

  • 1
    $\begingroup$ Since the antiderivative is linear, your answer would differ by at most a sign change. This would matter when you are plugging an antiderivate into, say, limits to calculate a definite integral. $\endgroup$ – amcalde Nov 17 '15 at 19:40

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










  • $\begingroup$ Thanks a lot for this answer, I just did not understand why you used the integral function, can't I just integrate indefinetly? $\endgroup$ – Gianolepo Nov 20 '15 at 16:56
  • $\begingroup$ @Francesco Caruso $\text{sgn}(2x+3)e^x$ is discontinuous at $x=-\dfrac{3}{2}$ , while $\text{sgn}(2x+3)\left(e^x-e^{-\frac{3}{2}}\right)$ is continuous at $x=-\dfrac{3}{2}$ . $\endgroup$ – Harry Peter Dec 4 '15 at 16:00

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