Question

1. $int \frac{1}{x^{2}-12x+37}dx$

Explanation:

Step1: Complete the square in denominator

$x^2-12x+37=(x-6)^2+1$

Step2: Substitute $u=x-6$, $du=dx$

$\int \frac{1}{u^2+1}du$

Step3: Apply arctangent integral rule

$\int \frac{1}{u^2+1}du=\arctan(u)+C$

Step4: Substitute back $u=x-6$

$\arctan(x-6)+C$

Answer:

$\arctan(x-6)+C$