# Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form

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## Contents

## Theorem

- $\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$

## Proof

\(\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} }\) | \(=\) | \(\displaystyle -\frac 1 a \csch^{-1} {\frac x a} + C\) | Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: $\csch^{-1}$ form | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\frac 1 a \map \ln {\frac {a + \sqrt{a^2 + x^2} } x} + C\) | $\csch^{-1} {\dfrac x a}$ in Logarithm Form |

$\blacksquare$

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: General Rules of Integration: $14.46$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.186$