What is the integral of xarctan(2/x) ?

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

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integral of xarctan(2/x)

\int x arctan (\frac{2}{x}) d x

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

Solution steps

\int x arctan (\frac{2}{x}) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (\frac{2}{x}) - \int - \frac{x ^{2}}{4 + x ^{2}} d x

= \frac{1}{2} x^{2} arctan (\frac{2}{x}) - \int - \frac{x ^{2}}{4 + x ^{2}} d x

\int - \frac{x ^{2}}{4 + x ^{2}} d x = 2 arctan (\frac{x}{2}) - x

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

Simplify

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

Add a constant to the solution

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

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