Primitive of Inverse Hyperbolic Tangent of x over a

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$\ds \int \tanh^{-1} \frac x a \rd x = x \tanh^{-1} \dfrac x a + \frac {a \map \ln {a^2 - x^2} } 2 + C$


With a view to expressing the primitive in the form:

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


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

and let:

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


\(\displaystyle \int \tanh^{-1} \frac x a \rd x\) \(=\) \(\displaystyle x \tanh^{-1} \frac x a - \int x \paren {\frac a {a^2 - x^2} } \rd x + C\) 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
\(\displaystyle \) \(=\) \(\displaystyle x \tanh^{-1} \frac x a - a \paren {-\frac 1 2 \map \ln {a^2 - x^2} } + C\) Primitive of $\dfrac x {a^2 - x^2}$
\(\displaystyle \) \(=\) \(\displaystyle x \tanh^{-1} \dfrac x a + \frac {a \map \ln {a^2 - x^2} } 2 + C\) simplifying


Also see