What is the integral of x^3arctan(x/2) ?

The integral of x^3arctan(x/2) is 1/4 x^4arctan(x/2)-4((x^3)/(24)-x/2+arctan(x/2))+C

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integral of x^3arctan(x/2)

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

\frac{1}{4} x^{4} arctan (\frac{x}{2}) - 4 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ^{4}}{2 ( x ^{2} + 4 )} d x = 4 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2}))

= \frac{1}{4} x^{4} arctan (\frac{x}{2}) - 4 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2}))

Add a constant to the solution

= \frac{1}{4} x^{4} arctan (\frac{x}{2}) - 4 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) + C

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What is the integral of x^3arctan(x/2) ?
The integral of x^3arctan(x/2) is 1/4 x^4arctan(x/2)-4((x^3)/(24)-x/2+arctan(x/2))+C