Primitive of x by Inverse Hyperbolic Sine of x over a

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Theorem

$\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$


Proof

With a view to expressing the primitive 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:

\(\ds u\) \(=\) \(\ds \arsinh \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 {\sqrt {x^2 + a^2} }\) Derivative of $\arsinh \dfrac x a$


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {x^2} 2\) Primitive of Power


Then:

\(\ds \int x \arsinh \frac x a \rd x\) \(=\) \(\ds \frac {x^2} 2 \arsinh \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^2} 2 \arsinh \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } + C\) simplifying
\(\ds \) \(=\) \(\ds \frac {x^2} 2 \arsinh \frac x a - \frac 1 2 \paren {\frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \arsinh \frac x a} + C\) Primitive of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
\(\ds \) \(=\) \(\ds \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C\) simplifying

$\blacksquare$


Also see


Sources

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