What is the integral of xarctan(20x) ?

The integral of xarctan(20x) is 1/2 x^2arctan(20x)-1/800 (-arctan(20x)+20x)+C

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integral of xarctan(20x)

\int x arctan (20 x) d x

\frac{1}{2} x^{2} arctan (20 x) - \frac{1}{800} (- arctan (20 x) + 20 x) + C

Solution steps

\int x arctan (20 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (20 x) - \int \frac{10 x ^{2}}{400 x ^{2} + 1} d x

= \frac{1}{2} x^{2} arctan (20 x) - \int \frac{10 x ^{2}}{400 x ^{2} + 1} d x

\int \frac{10 x ^{2}}{400 x ^{2} + 1} d x = \frac{1}{800} (- arctan (20 x) + 20 x)

= \frac{1}{2} x^{2} arctan (20 x) - \frac{1}{800} (- arctan (20 x) + 20 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (20 x) - \frac{1}{800} (- arctan (20 x) + 20 x) + C

\frac{x}{x ^{2} - 2}

\int (3 x^{2} + 5) cos (7 x) d x

x^{7} \cdot e^{x}

\frac{\partial}{\partial x} (x^{6} y^{9} - x^{5} y^{8})

\frac{\partial}{\partial x} (x - \frac{1}{x} e^{\frac{y}{x}})

What is the integral of xarctan(20x) ?
The integral of xarctan(20x) is 1/2 x^2arctan(20x)-1/800 (-arctan(20x)+20x)+C