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)

Add a constant to the solution

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

x \to 1 + lim (ln (ln (x)))

\frac{d}{d x} (e^{- 10 x})

\int (3 x + 2) (3 x^{2} + 4 x)^{4} d x

x \to - \infty lim (5^{x})

ln (\frac{( x + 2 )}{x - 2})

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