What is the integral of x*arctan(2x) ?

The integral of x*arctan(2x) is 1/2 x^2arctan(2x)-1/8 (-arctan(2x)+2x)+C

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integral of x*arctan(2x)

\int x \cdot arctan (2 x) d x

\frac{1}{2} x^{2} arctan (2 x) - \frac{1}{8} (- arctan (2 x) + 2 x) + C

Solution steps

\int x arctan (2 x) d x

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{4 x ^{2} + 1} d x = \frac{1}{8} (- arctan (2 x) + 2 x)

= \frac{1}{2} x^{2} arctan (2 x) - \frac{1}{8} (- arctan (2 x) + 2 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (2 x) - \frac{1}{8} (- arctan (2 x) + 2 x) + C

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What is the integral of x*arctan(2x) ?
The integral of x*arctan(2x) is 1/2 x^2arctan(2x)-1/8 (-arctan(2x)+2x)+C