Primitive of x by Inverse Hyperbolic Tangent of x over a

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Theorem

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


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:

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


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {x^2} 2\) Primitive of Power


Then:

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

$\blacksquare$


Also see


Sources