Evaluate the integral of x arctan x

by RoRi

November 30, 2015

Evaluate the following integral:

$\int x \arctan x \, dx.$

To evaluate this we use integration by parts, letting

$\begin{align*} u &= \arctan x & du &= \frac{1}{1+x^2} \, dx \\ dv &= x \, dx & dv &= \frac{x^2}{2}. \end{align*}$

Therefore,

$\begin{align*} \int x \arctan x \, dx &= \frac{x^2}{2} \arctan x - \frac{1}{2} \int \frac{x^2}{1+x^2} \, dx \\[9pt] &= \frac{1}{2} \left( x^2 \arctan x - \int \left(1 - \frac{1}{1+x^2} \right) \, dx \\[9pt] &= \frac{1}{2} \Big( (x^2+1)\arctan x - x \Big) + C. \end{align*}$

Stumbling Robot

Evaluate the integral of x arctan x

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