What is the integral of x*arctan(1/x) ?

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

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integral of x*arctan(1/x)

\int x \cdot arctan (\frac{1}{x}) d x

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

Solution steps

\int x arctan (\frac{1}{x}) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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