What is the integral of xarctan(2x) ?

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

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integral of xarctan(2x)

\int x 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

\frac{d}{d x} (\frac{1 - cos ( x )}{sin ( x )})

\frac{\partial}{\partial x} (y + \frac{x ^{2}}{y})

\frac{d}{d x} (\frac{1}{10} x^{5} - \frac{3}{4} x^{4} + \frac{4}{3} x^{3})

\frac{d}{d x} ((x^{2} + 1)^{3})

\frac{d}{d a} (1 + sin (a) + cos (a))

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