Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Proof 1

Theorem

$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$

for $0 < a < \size x$.

Proof

\(\ds \int \frac {\d x} {\sqrt {x^2 - a^2} }\)

\(=\)

\(\ds \cosh^{-1} {\frac x a} + C'\)

Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form

\(\ds \)

\(=\)

\(\ds \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} } + C'\)

Definition of Real Inverse Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } } + C'\)

rearrangement

\(\ds \)

\(=\)

\(\ds \map \ln {\frac {x + \sqrt {x^2 - a^2} } a} + C'\)

rearrangement

\(\ds \)

\(=\)

\(\ds \map \ln {x + \sqrt {x^2 - a^2} } - \ln a + C'\)

Difference of Logarithms

\(\ds \)

\(=\)

\(\ds \map \ln {x + \sqrt {x^2 - a^2} } + C\)

putting $C = -\ln a + C'$

$\blacksquare$