Primitives involving x squared plus a squared

Theorem

This page gathers together the primitives of some expressions involving $x^2 + a^2$.

Primitive of Reciprocal of $x^2 + a^2$: $\arctan$ Form

$\ds \int \frac {\d x} {x^2 + a^2} = \frac 1 a \arctan \frac x a + C$

Primitive of Reciprocal of $x^2 + a^2$: $\arccot$ Form

$\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$

Primitive $x$ over $x^2 + a^2$

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

Primitive $x^2$ over $x^2 + a^2$

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

Primitive $x^3$ over $x^2 + a^2$

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

Primitive of Reciprocal of $x \paren {x^2 + a^2}$

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

Primitive of Reciprocal of $x^2 \paren {x^2 + a^2}$

$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2} } = -\frac 1 {a^2 x} - \frac 1 {a^3} \arctan \frac x a + C$

Primitive of Reciprocal of $x^3 \paren {x^2 + a^2}$

$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2} } = -\frac 1 {2 a^2 x^2} - \frac 1 {2 a^4} \map \ln {\frac {x^2 + a^2} {x^2} } + C$