# Primitive of x by Inverse Hyperbolic Tangent of x over a

## 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

$\blacksquare$