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

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

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integral of x^2arctan(x/2)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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