Primitive of x by Inverse Hyperbolic Cotangent of x over a
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Theorem
- $\ds \int x \arcoth \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C$
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:
\(\ds u\) | \(=\) | \(\ds \arcoth \frac x a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac a {a^2 - x^2}\) | Derivative of $\arcoth \dfrac x a$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^2} 2\) | Primitive of Power |
Then:
\(\ds \int x \arcoth \frac x a \rd x\) | \(=\) | \(\ds \frac {x^2} 2 \arcoth \frac x a - \int \frac {x^2} 2 \paren {\frac a {a^2 - x^2} } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arcoth \frac x a + \frac a 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arcoth \frac x a + \frac a 2 \paren {x - a \arcoth \frac x a} + C\) | Primitive of $\dfrac {x^2} {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.662$