Answer
$2 \tan(2x+1) +C$
Work Step by Step
Our aim is to solve the integral $ \int (4x+2 ) \sec^2 (x^2+x) \ dx$
Let us consider that $a =x^2+x$ and $\dfrac{da}{dx}=2x+1 \implies dx=\dfrac{1}{2x+1} \ da$
Now, $ \int (4x+2 ) \sec^2 (x^2+x) \ dx = \int (4x+2 ) \sec^2 a (\dfrac{1}{2x+1}) \ da$
or, $= 2 \int \sec^2 a \ da+C$
or, $=2 \tan(2x+1) +C$
