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

## Theorem

$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$

## Proof

Let:

 $\displaystyle x$ $=$ $\displaystyle a \sinh \theta$ $\text {(1)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d x} {\d \theta}$ $=$ $\displaystyle a \cosh \theta$ Derivative of Hyperbolic Sine

Also:

 $\displaystyle x$ $=$ $\displaystyle a \sinh \theta$ $\displaystyle \leadsto \ \$ $\displaystyle x^2 + a^2$ $=$ $\displaystyle a^2 \sinh^2 \theta + a^2$ $\displaystyle$ $=$ $\displaystyle a^2 \paren {\sinh^2 \theta + 1}$ $\displaystyle$ $=$ $\displaystyle a^2 \cosh^2 \theta$ Difference of Squares of Hyperbolic Cosine and Sine $\text {(2)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle \sqrt {x^2 + a^2}$ $=$ $\displaystyle a \cosh \theta$

and:

 $\displaystyle x$ $=$ $\displaystyle a \sinh \theta$ $\text {(3)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle \theta$ $=$ $\displaystyle \sinh^{-1} \frac x a$ Definition 1 of Real Inverse Hyperbolic Sine

Thus:

 $\displaystyle \int \sqrt {x^2 + a^2} \rd x$ $=$ $\displaystyle \int \sqrt {x^2 + a^2} \, a \cosh \theta \rd \theta$ Integration by Substitution from $(1)$ $\displaystyle$ $=$ $\displaystyle \int a^2 \cosh^2 \theta \rd \theta$ substituting for $\sqrt {x^2 + a^2}$ from $(2)$ $\displaystyle$ $=$ $\displaystyle a^2 \int \cosh^2 \theta \rd \theta$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle a^2 \frac {\sinh \theta \cosh \theta + \theta} 2 + C$ Primitive of Square of Hyperbolic Cosine Function: Corollary $\displaystyle$ $=$ $\displaystyle \frac 1 2 a \sinh \theta a \cosh \theta + \frac {a^2 \theta} 2 + C$ rearranging $\displaystyle$ $=$ $\displaystyle \frac 1 2 x a \cosh \theta + \frac {a^2 \theta} 2 + C$ substituting $x = a \sinh \theta$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 x \sqrt {x^2 + a^2} + \frac {a^2 \theta} 2 + C$ substituting $\sqrt {x^2 + a^2} = a \cosh \theta$ from $(2)$ $\displaystyle$ $=$ $\displaystyle \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$ substituting for $\theta$ from $(3)$

$\blacksquare$