Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Sine Form

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

• Primitive of Reciprocal of $x \sqrt {x^2 - a^2}$
• Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$

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