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

## Theorem

For $a > 0$ and $0 < \size x < a$:

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

## Proof

 $\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} }$ $=$ $\ds -\frac 1 a \sech^{-1} {\frac {\size x} a} + C$ Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: $\sech^{-1}$ form $\ds$ $=$ $\ds -\frac 1 a \map \ln {\frac {1 + \sqrt {1 - \paren {\frac {\size x} a}^2} } {\frac {\size x} a} } + C$ Definition 2 of Inverse Hyperbolic Secant $\ds$ $=$ $\ds -\frac 1 a \map \ln {\frac {a + a \sqrt {1 - \paren {\frac {\size x} a}^2} } {\size x} } + C$ multiplying top and bottom by $a$ $\ds$ $=$ $\ds -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - a^2 \paren {\frac {\size x} a}^2} } {\size x} } + C$ moving $a$ within the square root $\ds$ $=$ $\ds -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ simplifying, noting $\size x^2 = x^2$

$\blacksquare$