Expand (x+2y)"2", to four terms.

Step 1: Use the double angle identity for sine, $\sin 2x = 2 \sin x \cos x$, to rewrite the integrand. $\int \frac{\sin 2x}{\cos x} \, dx = \int \frac{2 \sin x \cos x}{\cos x} \, dx$

Step 2: Simplify the integrand by canceling out $\cos x$. $\int 2 \sin x \, dx$

Step 3: Integrate the simplified expression. The integral of $\sin x$ is $-\cos x$. $2 \int \sin x \, dx = 2(-\cos x) + C$

Step 4: Write the final answer. $-2 \cos x + C$ The final answer is $\boxed{-2 \cos x + C}$.