# Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form

## Theorem

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

## Proof

Let:

 $\ds u$ $=$ $\ds \arsinh {\frac x a}$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \sinh u$ Definition of Real Area Hyperbolic Sine $\ds \leadsto \ \$ $\ds \frac {\d x} {\d u}$ $=$ $\ds a \cosh u$ Derivative of Hyperbolic Sine $\ds \leadsto \ \$ $\ds \int \frac {\d x} {\sqrt {x^2 + a^2} }$ $=$ $\ds \int \frac {a \cosh u} {\sqrt {a^2 \sinh^2 u + a^2} } \rd u$ Integration by Substitution $\ds$ $=$ $\ds \frac a a \int \frac {\cosh u} {\sqrt {\sinh^2 u + 1} } \rd u$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \int \frac {\cosh u} {\cosh u} \rd u$ Difference of Squares of Hyperbolic Cosine and Sine $\ds$ $=$ $\ds \int 1 \rd u$ $\ds$ $=$ $\ds u + C$ Integral of Constant $\ds$ $=$ $\ds \arsinh {\frac x a} + C$ Definition of $u$

$\blacksquare$