What is the integral of (x+1)arctan(x) ?

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

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integral of (x+1)arctan(x)

\int (x + 1) arctan (x) d x

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

Solution steps

\int (x + 1) arctan (x) d x

Expand

(x + 1) arctan (x) : x arctan (x) + arctan (x)

= \int x arctan (x) + arctan (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x arctan (x) d x + \int arctan (x) d x

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

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

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

Add a constant to the solution

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

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