What is the integral of zarctan(z) ?

The integral of zarctan(z) is 1/2 z^2arctan(z)-1/2 (-arctan(z)+z)+C

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integral of zarctan(z)

\int z arctan (z) d z

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

Solution steps

\int z arctan (z) d z

Apply Integration By Parts

: \frac{1}{2} z^{2} arctan (z) - \int \frac{z ^{2}}{2 ( z ^{2} + 1 )} d z

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

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

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

Add a constant to the solution

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

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