What's the best method to integrate $\int\arctan{\frac{1}{1 - x}}dx$?

Question

The only way I could think to integrate

$$\int\arctan{\frac{1}{1 - x}}dx$$

is by parts (g'(x) = dx), but the procedure is – I personally believe – way, way too tedious and time consuming for an exercise that is worth only 1/10 of the total mark.

Is there a better, faster way to integrate that expression? Thank you.

Answer 1

What you can do to shorten things is to write: $$I =\int \arctan\left(\frac1{1-x}\right)\, dx$$ as $$=-\int \operatorname{arccot} (x-1)\, dx$$

Now use the technique of integration by parts to get the required answer.

Answer 2

integrating by parts we obtain $$x\arctan\left(\frac{1}{1-x}\right)-\int\frac{x}{x^2-2x+2}dx$$

Answer 3

First substitute $1-x=t$, so the integral becomes $$ \int-\arctan\frac{1}{t}\,dt $$ Integrate by parts: $$ (-t)\arctan\frac{1}{t}-\int(-t)\frac{-1/t^2}{1+(1/t^2)}\,dt= -t\arctan\frac{1}{t}-\int\frac{t}{1+t^2}\,dt $$