What is the integral of xarctan(5x) ?

The integral of xarctan(5x) is 1/2 x^2arctan(5x)-1/50 (-arctan(5x)+5x)+C

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integral of xarctan(5x)

\int x arctan (5 x) d x

\frac{1}{2} x^{2} arctan (5 x) - \frac{1}{50} (- arctan (5 x) + 5 x) + C

Solution steps

\int x arctan (5 x) d x

Apply Integration By Parts

= \frac{1}{2} x^{2} arctan (5 x) - \int \frac{5 x ^{2}}{2 ( 25 x ^{2} + 1 )} d x

\int \frac{5 x ^{2}}{2 ( 25 x ^{2} + 1 )} d x = \frac{1}{50} (- arctan (5 x) + 5 x)

= \frac{1}{2} x^{2} arctan (5 x) - \frac{1}{50} (- arctan (5 x) + 5 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (5 x) - \frac{1}{50} (- arctan (5 x) + 5 x) + C

What is the integral of xarctan(5x) ?
The integral of xarctan(5x) is 1/2 x^2arctan(5x)-1/50 (-arctan(5x)+5x)+C