Answer
$$\int\cos2xdx=\frac{1}{2}\sin2x$$
Work Step by Step
To evaluate the integral $\int \cos2xdx$ we will use substitution $u=2x$ which gives us $du=2dx\Rightarrow \frac{1}{2}du=dx.$ Putting this into the integral we get:
$$\int\cos2xdx=\int\cos u\cdot \frac{1}{2}du=\frac{1}{2}\int\cos udu=\frac{1}{2}\sin u$$
Now we have to express solution in terms of $x$ by expressing $u$ in terms of $x$:
$$\int \cos2xdx=\frac{1}{2}\sin u=\frac{1}{2}\sin2x$$
