What is the integral of x^3*arctan(x) ?

The integral of x^3*arctan(x) is 1/4 x^4arctan(x)-1/4 ((x^3)/3-x+arctan(x))+C

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integral of x^3*arctan(x)

\int x^{3} \cdot arctan (x) d x

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

Solution steps

\int x^{3} arctan (x) d x

Apply Integration By Parts

= \frac{1}{4} x^{4} arctan (x) - \int \frac{x ^{4}}{4 ( x ^{2} + 1 )} d x

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

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

Add a constant to the solution

= \frac{1}{4} x^{4} arctan (x) - \frac{1}{4} (\frac{x ^{3}}{3} - x + arctan (x)) + C

What is the integral of x^3*arctan(x) ?
The integral of x^3*arctan(x) is 1/4 x^4arctan(x)-1/4 ((x^3)/3-x+arctan(x))+C