# Primitive of x cubed over a squared minus x squared

## Theorem

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

for $x^2 < a^2$.

## Proof

Let:

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

$\blacksquare$