Primitive of x by Inverse Hyperbolic Sine of x over a

Theorem

 * $\displaystyle \int x \sinh^{-1} \frac x a \ \mathrm d x = \left({\frac {x^2} 2 + \frac {a^2} 4}\right) \sinh^{-1} \frac x a + \frac {x \sqrt {x^2 + a^2} } 4 + C$

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 $x \cosh^{-1} \dfrac x a$


 * Primitive of $x \tanh^{-1} \dfrac x a$


 * Primitive of $x \coth^{-1} \dfrac x a$


 * Primitive of $x \operatorname{sech}^{-1} \dfrac x a$


 * Primitive of $x \operatorname{csch}^{-1} \dfrac x a$