The Integral of $\int \sin(ax) \cos(ax) dx$

Question

What is the integral of: $$I=\int \sin(ax) \cos(ax) dx$$

My approach is down below. I have attempted the problem and posted it as an answer. I did the problem using trigonometric substitution. $$u=ax$$ $$\frac{du}a=dx$$ $$\frac{1}a\int\sin u \cos u \ du$$ $$g=\sin u$$ $$dg = \cos u \ du$$ $$\frac{1}a\int g\ dg$$ $$I=\frac{1}a\frac{\sin^2ax}{2}+C$$

Answer 1

$$\sin(ax) \cos(ax)=\frac{1}{2}\sin(2ax)$$

Answer 2

$$u=ax$$ $$\frac{du}a=dx$$ $$\frac{1}a\int\sin u \cos u \ du$$ $$g=\sin u$$ $$dg = \cos u \ du$$ $$\frac{1}a\int g\ dg$$ $$I=\frac{1}a\frac{\sin^2ax}{2}+C$$