Primitive of x squared over Root of x squared minus a squared

Theorem

Inverse Hyperbolic Cosine Form

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

for $x > a$.

Logarithm Form

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

Also see

• Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
• Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$