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