sin(2x) − cos(2x) /(sin(2x) + cos(2x))^2  dx

To solve the integral 

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

we can use a substitution to simplify the expression.

### Step 1: Substitution
Let:
\[
u = \sin(2x) + \cos(2x)
\]
Then, the derivative \( du \) is:
\[
du = 2(\cos(2x) - \sin(2x)) \, dx
\]

Notice that the expression in the numerator, \( \sin(2x) - \cos(2x) \), is similar to the derivative of the substitution. We can rewrite the numerator:
\[
\sin(2x) - \cos(2x) = -\frac{1}{2} du
\]

### Step 2: Substitute and Simplify
Now, substitute into the integral:
\[
\int \frac{-\frac{1}{2} du}{u^2}
\]
This simplifies to:
\[
-\frac{1}{2} \int u^{-2} \, du
\]

### Step 3: Integrate
Now, integrate with respect to \( u \):
\[
-\frac{1}{2} \cdot \left( -\frac{1}{u} \right) + C = \frac{1}{2u} + C
\]

### Step 4: Back-Substitution
Finally, substitute back \( u = \sin(2x) + \cos(2x) \) to get:
\[
\frac{1}{2(\sin(2x) + \cos(2x))} + C
\]

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

Would you like more details or have any questions?

### Related Questions:
1. How can we apply substitution for different trigonometric integrals?
2. What other types of trigonometric identities are useful in integration?
3. Can we solve this integral using a different method?
4. How does this integral change if the power in the denominator is different?
5. What is the geometric interpretation of this integral?
6. How would the solution change if we replaced \(\sin(2x)\) with \(\sin(x)\)?
7. What happens if we integrate this expression with respect to a different variable?
8. How can we verify the result of this integral using differentiation?

### Tip:
When performing substitution in trigonometric integrals, always check if the derivative of the substitution closely matches any part of the integrand. This simplifies the process significantly.

Learn how to solve the integral of (sin(2x) - cos(2x)) / (sin(2x) + cos(2x))^2 with a detailed step-by-step solution using trigonometric substitution. This problem covers key concepts in trigonometric integrals and substitution methods. Suitable for advanced high school and college students.

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