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Evaluate the integral of x (arctan x)2

by RoRi

Evaluate the following integral

    \[ \int x (\arctan x)^2 \, dx. \]

We integrate by parts, letting

    \begin{align*} u &= \arctan x & du &= \frac{dx}{1+x^2} \\ dv &= x \arctan x \, dx & v &= \frac{1}{2} \Big( (x^2+1) \arctan x - x \Big). \end{align*}

(We know the integral of x \arctan x from this exercise, Section 6.22 Exercise #34.) Therefore, we have

    \begin{align*} \int x (\arctan x)^2 \, dx &= \frac{1}{2}(x^2+1)(\arctan x)^2 - \frac{x}{2} \arctan x - \frac{1}{2} \int \arctan x \, dx + \frac{1}{2} \int \frac{x}{1+x^2} \, dx \\[10pt] &= \frac{1}{2}(x^2+1)(\arctan x)^2 - \frac{x}{2} \arctan x \\ & \qquad - \frac{1}{2} \left( x \arctan x - \frac{1}{2} \log (1+x^2) \right) + \frac{1}{4} \log (1+x^2) + C \\[10pt] &= \frac{1}{2} (x^2+1)(\arctan x)^2 - x \arctan x + \frac{1}{2} \log (1+x^2) + C. \end{align*}

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