Question

q12 integration with completing the square
an antiderivative for $\frac{1}{x^2-2x+2}$ is

Explanation:

Step1: Complete the square in denominator

Rewrite $x^2-2x+2$ as $(x-1)^2+1$

Step2: Apply standard integral formula

Use $\int \frac{1}{u^2+1}du = \arctan(u) + C$, let $u=x-1$, $du=dx$
$\int \frac{1}{(x-1)^2+1}dx = \arctan(x-1) + C$

Answer:

$\arctan(x-1) + C$ (where $C$ is the constant of integration)