Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Sine Form
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Theorem
- $\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \sinh^{-1} {\frac a x} + C$
Proof
\(\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} }\) | \(=\) | \(\ds -\frac 1 a \csch^{-1} {\frac x a} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 a \sinh^{-1} {\frac a x} + C\) | Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant |
$\blacksquare$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals