What is the integral of xarctan(-x) ?

The integral of xarctan(-x) is 1/2 x^2arctan(-x)+1/2 (-arctan(x)+x)+C

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integral of xarctan(-x)

\int x arctan (- x) d x

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

Solution steps

\int x arctan (- x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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