What is the integral of uarctan(u) ?

The integral of uarctan(u) is 1/2 u^2arctan(u)-1/2 (-arctan(u)+u)+C

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integral of uarctan(u)

\int u arctan (u) d u

\frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u) + C

Solution steps

\int u arctan (u) d u

Apply Integration By Parts

= \frac{1}{2} u^{2} arctan (u) - \int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u

\int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u = \frac{1}{2} (- arctan (u) + u)

= \frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u)

Add a constant to the solution

= \frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u) + C

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What is the integral of uarctan(u) ?
The integral of uarctan(u) is 1/2 u^2arctan(u)-1/2 (-arctan(u)+u)+C