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

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

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integral of (x-1)arctan(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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