What is the integral of tarctan(t) ?

The integral of tarctan(t) is 1/2 t^2arctan(t)-1/2 (-arctan(t)+t)+C

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integral of tarctan(t)

\int t arctan (t) d t

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

Solution steps

\int t arctan (t) d t

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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