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

Theorem

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

With a view to expressing the problem in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

$\ds u$

$=$

$\ds x$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds 1$

and let:

$\ds \frac {\d v} {\d x}$

$=$

$\ds \frac x {\sqrt {x^2 + a^2} }$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds \sqrt {x^2 + a^2}$

Then:

$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} }$

$=$

$\ds \int x \frac {x \rd x} {\sqrt {x^2 + a^2} }$

$\ds $

$=$

$\ds x \sqrt {x^2 + a^2} - \int \sqrt {x^2 + a^2} \rd x$

$\ds $

$=$

$\ds x \sqrt {x^2 + a^2} - \left({\frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a}\right) + C$

$\ds $

$=$

$\ds \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$

$\blacksquare$