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

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

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integral of x^3arctan(x)

\int x^{3} 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

\frac{d}{d θ} (θ sin (θ))

x^{2} + 3 x - 2

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

- 0.1 x^{2} - x + 40

\int 7 x e^{(3 x - 1)} d x

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