# Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

## Theorem

$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \arcosh {\size {\frac x a} } + C$

for $x^2 > a^2$.

## Proof

When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and then $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.

When $\size x < a$ we have that $x^2 - a^2 < 0$ and then $\sqrt {x^2 - a^2}$ is not defined.

Hence the domain needs to be restricted to $\size x > a$, or that is: $\size {\dfrac x a} > 1$.

Let $x > a$.

Let:

 $\ds u$ $=$ $\ds \map \arcosh {\frac x a}$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \cosh u$ $\ds \leadsto \ \$ $\ds \frac {\d x} {\d u}$ $=$ $\ds a \sinh u$ Derivative of Hyperbolic Cosine Function $\ds \leadsto \ \$ $\ds \int \frac {\d x} {\sqrt {x^2 - a^2} }$ $=$ $\ds \int \frac {a \sinh u} {\sqrt {a^2 \cosh^2 u - a^2} } \rd u$ Integration by Substitution $\ds$ $=$ $\ds \frac a a \int \frac {\sinh u} {\sqrt {\cosh^2 u - 1} } \rd u$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \int \frac {\sinh u} {\sinh u} \rd u$ Difference of Squares of Hyperbolic Cosine and Sine $\ds$ $=$ $\ds \int 1 \rd u$ $\ds$ $=$ $\ds u + C$ Integral of Constant $\ds$ $=$ $\ds \map \arcosh {\frac x a} + C$ Definition of $u$ $\ds$ $=$ $\ds \arcosh \size {\frac x a} + C$ as $\dfrac x a > 0$ $\ds$ $=$ $\ds \dfrac {\size x} x \arcosh \size {\frac x a} + C$ as $x > 0$, so $\dfrac {\size x} x = 1$

Let $x < -a$.

Let $z = -x$.

Hence:

$\dfrac {\d z} {\d x} = -1$

Then:

 $\ds \int \frac {\d x} {\sqrt {x^2 - a^2} }$ $=$ $\ds \int \frac {-\d z} {\sqrt {\paren {-z}^2 - a^2} }$ Integration by Substitution $\ds$ $=$ $\ds -\int \frac {\d z} {\sqrt {z^2 - a^2} }$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds -\map \arcosh {\frac z a} + C$ from above $\ds$ $=$ $\ds -\map \arcosh {\frac {-x} a} + C$ substituting back $\ds$ $=$ $\ds -\arcosh \size {\frac x a} + C$ as $\dfrac x a < 0$ $\ds$ $=$ $\ds \dfrac {\size x} x \arcosh \size {\frac x a} + C$ as $x < 0$, so $\dfrac {\size x} x = -1$

$\blacksquare$

## Sources

but beware that he glosses over the details of what happens for negative $x$.
who does the same for negative $x$.
who covers positive $x$ only