Integral of $\sin(ax+b)\cos(ax+b)$

Question

$\displaystyle\int \sin(ax+b)\cos(ax+b) \, dx$ is giving me two answers;

1. $\dfrac{1}{2a}\sin^2(ax+b)+C$

2. negative of $\dfrac{1}{4a}\cos2(ax+b) +C$

Are these two results the same? Can you please prove it? Thank you

Answer 1

They are the same because you add a fixed constant to an arbitrary constant you should get back an arbitrary constant.

Answer 2

They are same. Just use the formula $$\cos 2A = 1 - 2 \sin^2 A$$