Primitive of x squared by Inverse Hyperbolic Cosine of x over a
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Theorem
- $\ds \int x^2 \arcosh \frac x a \rd x = \dfrac {x^3} 3 \arcosh \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$
where $\arcosh$ denotes the real area hyperbolic cosine.
Corollary
- $\ds \int x^2 \paren {-\cosh^{-1} \dfrac x a} \rd x = \dfrac {x^3} 3 \paren {-\cosh^{-1} \dfrac x a} - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
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:
\(\ds u\) | \(=\) | \(\ds \arcosh \frac x a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac 1 {\sqrt {x^2 - a^2} }\) | Derivative of $\arcosh \dfrac x a$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^3} 3\) | Primitive of Power |
Then:
\(\ds \int x^2 \arcosh \frac x a \rd x\) | \(=\) | \(\ds \frac {x^3} 3 \arcosh \frac x a - \int \frac {x^3} 3 \paren {\frac 1 {\sqrt {x^2 - a^2} } } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^3} 3 \arcosh \frac x a - \frac 1 3 \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^3} 3 \arcosh \frac x a - \frac 1 3 \paren {\frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} } + C\) | Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^3} 3 \arcosh \frac x a - \frac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.653$