Home » Blog » Evaluate the integral of x arctan x

Evaluate the integral of x arctan x

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*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):