Primitive of x squared by Inverse Hyperbolic Cosine of x over a

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Theorem

$\displaystyle \int x^2 \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a > 0 \\ \displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a < 0 \end{cases}$


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:

\(\displaystyle u\) \(=\) \(\displaystyle \cosh^{-1} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle \frac 1 {\sqrt {x^2 - a^2} }\) Derivative of $\cosh^{-1} \dfrac x a$


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle x^2\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {x^3} 3\) Primitive of Power


Then:

\(\displaystyle \int x^2 \cosh^{-1} \frac x a \ \mathrm d x\) \(=\) \(\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \int \frac {x^3 \ \mathrm d x} {\sqrt {x^2 - a^2} } + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \left({\frac {\left({\sqrt {x^2 - a^2} }\right)^3} 3 + a^2 \sqrt {x^2 - a^2} }\right) + C\) Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C\) simplifying

$\blacksquare$



Also see


Sources