To find the indefinite integral of \( \sin(2x) \), we proceed as follows:

\[
\int \sin(2x) \, dx
\]

1. Use a substitution to simplify the integral. Let \( u = 2x \), then \( du = 2 \, dx \) or \( dx = \frac{du}{2} \).

2. Substitute into the integral:

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

3. Now integrate \( \sin(u) \):

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

4. Substitute \( u = 2x \) back into the result:

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

Thus, the answer is:

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

Do you want more details on the substitution method or have any other questions?

### Related Questions
1. What is the indefinite integral of \( \cos(2x) \)?
2. How do you find the indefinite integral of \( \sin(kx) \) for any constant \( k \)?
3. What is the integral of \( \sin^2(2x) \)?
4. How would you solve \( \int \cos^2(2x) \, dx \)?
5. What is the definite integral of \( \sin(2x) \) from \( 0 \) to \( \pi \)?

**Tip:** Remember that substitution is a powerful technique for integrals involving functions of \( ax + b \), where \( a \) is a constant.

Learn how to solve the indefinite integral of sin(2x) using substitution. This detailed solution involves key integration techniques and trigonometric identities. Suitable for students in Grades 11-12 studying integration.

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