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

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Theorem

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

for $x > a$.


Proof

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:

\(\ds u\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Power Rule for Derivatives


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac x {\sqrt {x^2 - a^2} }\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \sqrt {x^2 - a^2}\) Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$


Then:

\(\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} }\) \(=\) \(\ds \int x \frac {x \rd x} {\sqrt {x^2 - a^2} }\)
\(\ds \) \(=\) \(\ds x \sqrt {x^2 - a^2} - \int \sqrt {x^2 - a^2} \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds x \sqrt {x^2 - a^2} - \paren {\frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a} + C\) Primitive of $\sqrt {x^2 - a^2}$
\(\ds \) \(=\) \(\ds \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C\) simplifying

Note that because:

$\cosh^{-1} \dfrac x a$ is defined for $x \ge a$ only

and:

$\dfrac {x^2} {\sqrt {x^2 - a^2} }$ is not defined for $x = a$

$x$ is constrained as indicated.

$\blacksquare$


Also see


Sources

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