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

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Theorem

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


Proof

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

$\blacksquare$


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