What is the integral of 4x^2arctan(x) ?

The integral of 4x^2arctan(x) is 4(1/3 x^3arctan(x)-1/6 (x^2+1-ln|x^2+1|))+C

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integral of 4x^2arctan(x)

\int 4 x^{2} arctan (x) d x

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

Solution steps

\int 4 x^{2} arctan (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x^{2} arctan (x) d x

Apply Integration By Parts

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

= 4 (\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)

= 4 (\frac{1}{3} x^{3} arctan (x) - \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1))

Add a constant to the solution

= 4 (\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 4x^2arctan(x) ?
The integral of 4x^2arctan(x) is 4(1/3 x^3arctan(x)-1/6 (x^2+1-ln|x^2+1|))+C