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

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

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integral of xarctan(x+1)

\int x arctan (x + 1) d x

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

Solution steps

\int x arctan (x + 1) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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