What is the integral of-tarctan(t) ?

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

Solution steps

\int - t arctan (t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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))

Simplify

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

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

Add a constant to the solution

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