Primitive of x by Inverse Hyperbolic Secant of x over a
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Theorem
- $\ds \int x \arsech \frac x a \rd x = \dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $\arsech$ denotes the real area hyperbolic secant.
Corollary
- $\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant 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:
\(\ds u\) | \(=\) | \(\ds \arsech \frac x a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {-a} {x \sqrt {a^2 - x^2} }\) | Derivative of $\arsech \dfrac x a$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^2} 2\) | Primitive of Power |
Then:
\(\ds \int x \arsech \frac x a \rd x\) | \(=\) | \(\ds \frac {x^2} 2 \arsech \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x \sqrt {a^2 - x^2} } } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arsech \frac x a + \frac a 2 \int \frac {x \rd x} {\sqrt {a^2 - x^2} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arsech \frac x a + \frac a 2 \paren {-\sqrt {a^2 - x^2} } + C\) | Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arsech \frac x a - \frac {a \sqrt {a^2 - x^2} } 2 + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.667$