Stop using this u-substitution! Integrate cos(2x) by using the chain rule backwards.
I implore you to stop using this u-substitution! For the integral of cos(2x), we can use an explicit u-substitution: let u=2x, du=2dx and dx=1/2du. We transform the integral to u-space to get 1/2*integral(cos(u))du, which integrates to 1/2*sin(u)+C or 1/2*sin(2x)+C.
But that's WAY too much machinery for such a simple integral! We simply want the antiderivative of cos(2x), which is essentially sin(2x). Test it out: the derivative of sin(2x) is 2cos(2x) because the chain rule produces a factor of 2. All we have to do is provide a factor of 1/2 out in front to take care of that factor of 2, and we have the answer with a simple guess and adjust method: 1/2*sin(2x)+C.
As you move into more advanced courses in math, physics and engineering, you will have to compute this sort of integral repeatedly and casually, and students are expected to just quickly guess and adjust to write down the answer by using the chain rule backwards, rather than using an explicit u-substitution.
Hopefully this video gives you the confidence to throw the trivial u-substitution out of your tool kit, and go forth happily guessing and adjusting to compute integrals of simple function compositions!
