To find the indefinite integral of the function $\sin(2x) - \cos(2x) \, dx$ using integration by substitution, we will handle each term separately.

Step 1: Integrate $\sin(2x) \, dx$

Let's use the substitution method for the first term.

1. Substitute: Let $u = 2x$, hence $du = 2 \, dx$ 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. Integrate: The integral of $\sin(u)$ with respect to $u$ is $-\cos(u)$: $\frac{1}{2} \int \sin(u) \, du = -\frac{1}{2} \cos(u)$

4. Substitute back $u = 2x$: $-\frac{1}{2} \cos(2x)$

Step 2: Integrate $\cos(2x) \, dx$

1. Substitute: Let $u = 2x$, so $du = 2 \, dx$ or $dx = \frac{du}{2}$.

2. Substitute into the integral: $\int \cos(2x) \, dx = \int \cos(u) \cdot \frac{du}{2} = \frac{1}{2} \int \cos(u) \, du$

3. Integrate: The integral of $\cos(u)$ with respect to $u$ is $\sin(u)$: $\frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u)$

4. Substitute back $u = 2x$: $\frac{1}{2} \sin(2x)$

Step 3: Combine the results

Now, we combine the integrals for both terms:

$\int \left(\sin(2x) - \cos(2x)\right) \, dx = -\frac{1}{2} \cos(2x) + \frac{1}{2} \sin(2x) + C$

Final Answer:

$\int \left(\sin(2x) - \cos(2x)\right) \, dx = \frac{1}{2} \sin(2x) - \frac{1}{2} \cos(2x) + C$

This is the indefinite integral of the function using integration by substitution.

Do you want any details or have any questions? Here are 8 related questions you might find interesting:

1. How do you choose the right substitution in integration?
2. What are common trigonometric identities useful in integration?
3. How does substitution simplify trigonometric integrals?
4. What are the advantages of using substitution over direct integration?
5. Can substitution be used for definite integrals, and how?
6. What is the integral of other common trigonometric functions like $\tan(x)$ or $\sec(x)$?
7. How do you reverse the substitution after integrating?
8. What is the difference between substitution and integration by parts?

Tip: When using substitution in trigonometric integrals, always check if the integral can be simplified further by using trigonometric identities.