What is the integral of (x+4)arctan(x/2) ?

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

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integral of (x+4)arctan(x/2)

\int (x + 4) arctan (\frac{x}{2}) d x

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

Solution steps

\int (x + 4) arctan (\frac{x}{2}) d x

Apply Integration By Parts

: arctan (\frac{x}{2}) (\frac{x ^{2}}{2} + 4 x) - \int \frac{x ^{2} + 8 x}{x ^{2} + 4} d x

= arctan (\frac{x}{2}) (\frac{x ^{2}}{2} + 4 x) - \int \frac{x ^{2} + 8 x}{x ^{2} + 4} d x

\int \frac{x ^{2} + 8 x}{x ^{2} + 4} d x = - 2 arctan (\frac{x}{2}) + x + 4 ln x^{2} + 4

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

Simplify

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

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

Add a constant to the solution

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

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