What is the integral of xarctan(3x) ?

The integral of xarctan(3x) is 1/2 x^2arctan(3x)-1/18 (-arctan(3x)+3x)+C

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integral of xarctan(3x)

\int x arctan (3 x) d x

\frac{1}{2} x^{2} arctan (3 x) - \frac{1}{18} (- arctan (3 x) + 3 x) + C

Solution steps

\int x arctan (3 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (3 x) - \int \frac{3 x ^{2}}{2 ( 9 x ^{2} + 1 )} d x

= \frac{1}{2} x^{2} arctan (3 x) - \int \frac{3 x ^{2}}{2 ( 9 x ^{2} + 1 )} d x

\int \frac{3 x ^{2}}{2 ( 9 x ^{2} + 1 )} d x = \frac{1}{18} (- arctan (3 x) + 3 x)

= \frac{1}{2} x^{2} arctan (3 x) - \frac{1}{18} (- arctan (3 x) + 3 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (3 x) - \frac{1}{18} (- arctan (3 x) + 3 x) + C

x \to 0 - lim (\frac{∣ x - 7 ∣}{x - 7})

y^{^{''}} + 8 y^{^{'}} + 16 y = 0, y (0) = 3, y^{^{'}} (0) = 8

8 3 x^{2} + 9

\frac{\partial}{\partial x} (42 e^{7 x} y)

f (x) = \frac{x}{x ^{2} + 64}

What is the integral of xarctan(3x) ?
The integral of xarctan(3x) is 1/2 x^2arctan(3x)-1/18 (-arctan(3x)+3x)+C