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

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

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integral of arctan(x-1)

\int arctan (x - 1) d x

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

Solution steps

\int arctan (x - 1) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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