# Primitive of x cubed over x squared minus a squared

## Theorem

$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) + C$

for $x^2 > a^2$.

## Proof

Let:

 $\displaystyle \int \frac {x^3 \ \mathrm d x} {x^2 - a^2}$ $=$ $\displaystyle \int \frac {x \left({x^2 - a^2 + a^2}\right)} {x^2 - a^2} \ \mathrm d x$ $\displaystyle$ $=$ $\displaystyle \int \frac {x \left({x^2 - a^2}\right)} {x^2 - a^2} \ \mathrm d x + \int \frac {a^2 x} {x^2 - a^2} \ \mathrm d x$ $\displaystyle$ $=$ $\displaystyle \int x \ \mathrm d x + a^2 \int \frac {x \ \mathrm d x} {x^2 - a^2}$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 + a^2 \int \frac {x \ \mathrm d x} {x^2 - a^2} + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 + a^2 \left({\frac 1 2 \ln \left({x^2 - a^2}\right)}\right) + C$ Primitive of $\dfrac x {x^2 - a^2}$ $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) + C$ simplifying

$\blacksquare$