What is the integral of x*arctan(7x) ?

The integral of x*arctan(7x) is 1/2 x^2arctan(7x)-1/98 (-arctan(7x)+7x)+C

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integral of x*arctan(7x)

\int x \cdot arctan (7 x) d x

\frac{1}{2} x^{2} arctan (7 x) - \frac{1}{98} (- arctan (7 x) + 7 x) + C

Solution steps

\int x arctan (7 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (7 x) - \int \frac{7 x ^{2}}{2 ( 49 x ^{2} + 1 )} d x

= \frac{1}{2} x^{2} arctan (7 x) - \int \frac{7 x ^{2}}{2 ( 49 x ^{2} + 1 )} d x

\int \frac{7 x ^{2}}{2 ( 49 x ^{2} + 1 )} d x = \frac{1}{98} (- arctan (7 x) + 7 x)

= \frac{1}{2} x^{2} arctan (7 x) - \frac{1}{98} (- arctan (7 x) + 7 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (7 x) - \frac{1}{98} (- arctan (7 x) + 7 x) + C

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What is the integral of x*arctan(7x) ?
The integral of x*arctan(7x) is 1/2 x^2arctan(7x)-1/98 (-arctan(7x)+7x)+C