What is the integral of xarctan(x)) ?

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

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integral of xarctan(x))

\int x d (arctan (x)) d x

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

Solution steps

\int x d arctan (x) d x

Take the constant out:

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

= d \cdot \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

= d (\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)

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

Add a constant to the solution

= d (\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 d(1/2 x^2arctan(x)-1/2 (-arctan(x)+x))+C