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

\int_{0}^{t} t^{2} e^{- 4 t} d t

\frac{d}{d x} (x + tan (x))

\int x^{6} e^{x} d x

(4 x^{2} - 8 x + 8) e^{x}

4 y^{^{''}} + 11 y = 0

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