Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form

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Theorem

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

where $x^2 < a^2$.


Proof

Let:

\(\displaystyle \forall x \in \R: \size {\frac x a} < 1: \ \ \) \(\displaystyle u\) \(=\) \(\displaystyle \tanh^{-1} {\frac x a}\)
\(\displaystyle \forall x \in \R: x^2 < a^2: \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle a \tanh u\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall x \in \R: x^2 < a^2: \ \ \) \(\displaystyle \frac {\d x} {\d u}\) \(=\) \(\displaystyle a \sech^2 u\) Derivative of Hyperbolic Tangent Function
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall x \in \R: x^2 < a^2: \ \ \) \(\displaystyle \int \frac 1 {a^2 - x^2} \rd x\) \(=\) \(\displaystyle \int \frac {a \sech^2 u} {a^2 - a^2 \tanh^2 u} \rd u\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac a {a^2} \int \frac {\sech^2 u} {1 - \tanh^2 u} \rd u\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \frac {\sech^2 u} {\sech^2 u} \rd u\) Sum of Squares of Hyperbolic Secant and Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \rd u\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a u + C\) Integral of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \tanh^{-1} \frac x a + C\) Definition of $u$

$\blacksquare$


Also see


Sources