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

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

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integral of arctan(x)x^2

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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