# Primitive of x cubed over x squared plus a squared

## Theorem

$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$

## Proof

 $\ds \int \frac {x^3 \rd x} {x^2 + a^2}$ $=$ $\ds \int \paren {x - \frac {a^2 x} {x^2 + a^2} } \rd x$ long division $\ds$ $=$ $\ds \int x \rd x - a^2 \int \frac {x \rd x} {x^2 + a^2}$ Linear Combination of Integrals $\ds$ $=$ $\ds \frac {x^2} 2 - a^2 \int \frac {x \rd x} {x^2 + a^2} + C$ Primitive of Power $\ds$ $=$ $\ds \frac {x^2} 2 - a^2 \paren {\frac 1 2 \map \ln {x^2 + a^2} } + C$ Primitive of $\dfrac x {x^2 + a^2}$ $\ds$ $=$ $\ds \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ simplifying

$\blacksquare$