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

= \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

\int (\frac{x ^{2}}{3} - \frac{3}{2 x - 3} - e^{3 x}) d x

\int x (5 - 2 x)^{\frac{1}{3}} d x

\int \frac{x ^{2}}{( a ^{2} + x ^{2} ) ^{2}} d x

\int \frac{e ^{2 w}}{1 - e ^{2 w}} d w

\frac{2 x}{9 x ^{2} - 6 x - 8}

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