What is the integral of x^2arctan(6x) ?

The integral of x^2arctan(6x) is 1/3 x^3arctan(6x)-1/1296 (36x^2+1-ln|36x^2+1|)+C

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integral of x^2arctan(6x)

\int x^{2} arctan (6 x) d x

\frac{1}{3} x^{3} arctan (6 x) - \frac{1}{1296} (36 x^{2} + 1 - ln 36 x^{2} + 1) + C

Solution steps

\int x^{2} arctan (6 x) d x

Apply Integration By Parts

: \frac{1}{3} x^{3} arctan (6 x) - \int \frac{2 x ^{3}}{36 x ^{2} + 1} d x

= \frac{1}{3} x^{3} arctan (6 x) - \int \frac{2 x ^{3}}{36 x ^{2} + 1} d x

\int \frac{2 x ^{3}}{36 x ^{2} + 1} d x = \frac{1}{1296} (36 x^{2} + 1 - ln 36 x^{2} + 1)

= \frac{1}{3} x^{3} arctan (6 x) - \frac{1}{1296} (36 x^{2} + 1 - ln 36 x^{2} + 1)

Add a constant to the solution

= \frac{1}{3} x^{3} arctan (6 x) - \frac{1}{1296} (36 x^{2} + 1 - ln 36 x^{2} + 1) + C

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\frac{\partial}{\partial u} (2 v cos (u))

What is the integral of x^2arctan(6x) ?
The integral of x^2arctan(6x) is 1/3 x^3arctan(6x)-1/1296 (36x^2+1-ln|36x^2+1|)+C