# Primitive of x cubed over x squared plus a squared squared

## Theorem

$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$

## Proof 1

Let:

 $\ds z$ $=$ $\ds x^2 + a^2$ $\ds \leadsto \ \$ $\ds \frac {\d z} {\d x}$ $=$ $\ds 2 x$ Derivative of Power $\ds \leadsto \ \$ $\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}$ $=$ $\ds \int \frac {\paren {z - a^2} } {z^2} \frac {\d z} 2$ Integration by Substitution $\ds$ $=$ $\ds \frac 1 2 \int \frac {\d z} z - \frac {a^2} 2 \int \frac {\d z} {z^2}$ Linear Combination of Primitives $\ds$ $=$ $\ds \frac 1 2 \int \frac {\d z} z + \frac {a^2} {2 z} + C$ Primitive of Power $\ds$ $=$ $\ds \frac 1 2 \ln z + \frac {a^2} {2 z} + C$ Primitive of Reciprocal $\ds$ $=$ $\ds \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$ substituting for $z$

$\blacksquare$

## Proof 2

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

$\blacksquare$