Answer
We can express the most general antiderivative $F(x)$ of the function as follows:
$F(x) = -\frac{1}{2}x^{-2}+ sinh~x+C_1~~~~$ if $x \gt 0$
$F(x) = -\frac{1}{2}x^{-2}+ sinh~x+C_2~~~~$ if $x \lt 0$
Work Step by Step
We can find the most general antiderivative $F(x)$ of the function:
$\int~f(x)$
$= \int (x^{-3}+ cosh~x)$
$= \int x^{-3}+ \int cosh~x$
$= \int x^{-3}+ \int \frac{e^x+e^{-x}}{2}$
$= \int x^{-3}+ \int \frac{e^x}{2}+ \int \frac{e^{-x}}{2}$
$= -\frac{1}{2}x^{-2}+ \frac{e^x}{2}- \frac{e^{-x}}{2}+C$
$= -\frac{1}{2}x^{-2}+ sinh~x+C$
Note that $-\frac{1}{2}x^{-2}$ does not exist when $x=0$
We can express the most general antiderivative $F(x)$ of the function as follows:
$F(x) = -\frac{1}{2}x^{-2}+ sinh~x+C_1~~~~$ if $x \gt 0$
$F(x) = -\frac{1}{2}x^{-2}+ sinh~x+C_2~~~~$ if $x \lt 0$
