Primitive of Reciprocal of Root of x squared plus a squared

Theorem

Inverse Hyperbolic Sine Form

$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \arsinh {\frac x a} + C$

Logarithm Form

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

Examples

Primitive of $\dfrac 1 {\sqrt {16 + 9 x^2} }$

$\ds \int \dfrac 1 {\sqrt {16 + 9 x^2} } = \dfrac 1 3 \map \ln {3 x + \sqrt {9 x^2 + 16} }$

Also see

• Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$
• Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$