To integrate \(\sin(2x)\), we can use a substitution method. Let \(u = 2x\). Then, \(du = 2dx\) or \(dx = \frac{1}{2}du\).

Now, we can rewrite the integral in terms of \(u\):

\[
\int \sin(2x) \, dx = \int \sin(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int \sin(u) \, du
\]

The integral of \(\sin(u)\) is \(-\cos(u)\). So, we have:

\[
\frac{1}{2} \int \sin(u) \, du = \frac{1}{2} (-\cos(u)) = -\frac{1}{2} \cos(u)
\]

Now, substitute back \(u = 2x\):

\[
-\frac{1}{2} \cos(2x) + C
\]

Therefore, the integral of \(\sin(2x)\) is:

\[
\int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C
\]

Would you like more details on this integration or have any other questions?

Here are 5 questions you might consider asking:
1. How do you integrate \(\cos(2x)\)?
2. What is the integral of \(\tan(x)\)?
3. How do you perform integration by substitution?
4. What is the integral of \(\sin^2(x)\)?
5. Can you explain how to integrate \(\sin(ax)\) for any constant \(a\)?

**Tip:** When dealing with integrals involving trigonometric functions with coefficients inside the argument (like \(\sin(2x)\)), substitution is a powerful method to simplify the integration process.

Learn how to integrate sin(2x) using the substitution method. Step-by-step solution with detailed explanations of the substitution process and integration of trigonometric functions. Suitable for advanced high school students and anyone studying calculus.

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