How to integrate $|x^n|$

Question

How to integrate $|x^n|$? The answer was given as $\frac{|x^n|x}{n+1}$. How do you find this? I would also like to know how do find the anti derivative of a modulo function when interval is not given.

Answer 1

First consider the positive semi-axis and use the fact that the absolute value can be omitted on that subset. Then consider the negative semi-axis and use $|x|=-x$ on that subset. Now see that your answers can be unified using the proposed solution.

Answer 2

Case $1$: $n\neq-1$

$\int|x^n|~dx$

$=\int x^n\text{sgn}(x^n)~dx$

$=\dfrac{x^{n+1}\text{sgn}(x^n)}{n+1}+C$

$=\dfrac{|x^n|x}{n+1}+C$

Case $2$: $n=-1$

$\int\left|\dfrac{1}{x}\right|~dx$

$=\int\text{sgn}\left(\dfrac{1}{x}\right)\dfrac{1}{x}~dx$

$=\text{sgn}\left(\dfrac{1}{x}\right)\ln x+C$