Primitive of Inverse Hyperbolic Sine of x over a

Theorem

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

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then:

Also see

 * Primitive of $\cosh^{-1} \dfrac x a$
 * Primitive of $\tanh^{-1} \dfrac x a$
 * Primitive of $\coth^{-1} \dfrac x a$
 * Primitive of $\operatorname{sech}^{-1} \dfrac x a$
 * Primitive of $\operatorname{csch}^{-1} \dfrac x a$