Primitive of Reciprocal of x squared plus a squared

Theorem

Arctangent Form

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

Arccotangent Form

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

Examples

Primitive of $\dfrac 1 {16 + x^2}$

$\ds \int \dfrac 1 {16 + x^2} \rd x = \dfrac 1 4 \arctan \dfrac x 4 + C$