What is the integral of arctan(x)*x ?

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

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integral of arctan(x)*x

\int arctan (x) \cdot x d x

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

Solution steps

\int arctan (x) 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)

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