Primitive of Reciprocal of x by Root of a squared minus x squared
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Theorem
Inverse Hyperbolic Secant Form
For $a > 0$ and $0 < \size x < a$:
- $\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C$
Logarithm Form
For $a > 0$ and $0 < \size x < a$:
- $\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$