Primitive of Inverse Hyperbolic Tangent of x over a

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Theorem

$\displaystyle \int \tanh^{-1} \frac x a \rd x = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C$


Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\displaystyle u\) \(=\) \(\displaystyle \tanh^{-1} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle \frac a {a^2 - x^2}\) Derivative of $\tanh^{-1} \dfrac x a$


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle x\) Primitive of Constant


Then:

\(\displaystyle \int \tanh^{-1} \frac x a \rd x\) \(=\) \(\displaystyle x \tanh^{-1} \frac x a - \int x \left({\frac a {a^2 - x^2} }\right) \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle x \tanh^{-1} \frac x a - a \int \frac {x \rd x} {a^2 - x^2} + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle x \tanh^{-1} \frac x a - a \left({- \frac 1 2 \ln \left({a^2 - x^2}\right)}\right) + C\) Primitive of $\dfrac x {a^2 - x^2}$
\(\displaystyle \) \(=\) \(\displaystyle x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C\) simplifying

$\blacksquare$


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