Primitive of x by Inverse Hyperbolic Cosecant of x over a
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Theorem
- $\ds \int x \arcsch \frac x a \rd x = \begin {cases} \dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0 \\ \end {cases}$
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:
\(\ds u\) | \(=\) | \(\ds \arcsch \frac x a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \dfrac {-a} {\size x \sqrt {a^2 + x^2} }\) | Derivative of $\arcsch \dfrac x a$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^2} 2\) | Primitive of Power |
Then:
\(\ds \int x \arcsch \frac x a \rd x\) | \(=\) | \(\ds \frac {x^2} 2 \arcsch \frac x a - \int \frac {x^2} 2 \paren {\dfrac {-a} {\size x \sqrt {a^2 + x^2} } } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arcsch \frac x a + \frac a 2 \int \frac {x^2 \rd x} {\size x \sqrt {a^2 + x^2} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arcsch \frac x a \begin {cases} \mathop + \dfrac a 2 \int \dfrac {x \rd x} {\sqrt {x^2 + a^2} } + C & : x > 0 \\ \mathop - \dfrac a 2 \int \dfrac {x \rd x} {\sqrt {x^2 + a^2} } + C & : x < 0 \end {cases}\) | Definition of Absolute Value | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arcsch \frac x a \begin {cases} \mathop + \dfrac a 2 \sqrt {x^2 + a^2} + C & : x > 0 \\ \mathop - \dfrac a 2 \sqrt {x^2 + a^2} + C & : x < 0 \end {cases}\) | Primitive of $\dfrac x {\sqrt {x^2 + a^2} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} \dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0 \\ \end {cases}\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.670$