Solve the following problem: ∫1/sin(2x)^3

To solve the integral \(\int \frac{1}{\sin^3(2x)} \, dx\), we'll use a trigonometric identity and substitution.

### Step 1: Use Trigonometric Identity
We can express \(\sin^3(2x)\) as \(\sin(2x)\) times \(\sin^2(2x)\). Using the identity \(\sin^2(2x) = 1 - \cos^2(2x)\), we can rewrite the integral as:

\[
\int \frac{1}{\sin^3(2x)} \, dx = \int \frac{1}{\sin(2x) \cdot \sin^2(2x)} \, dx = \int \frac{1}{\sin(2x) \cdot (1 - \cos^2(2x))} \, dx
\]

### Step 2: Use the Substitution Method
Let \( u = \cos(2x) \), so that \( du = -2\sin(2x) \, dx \), or \( \frac{du}{-2} = \sin(2x) \, dx \).

Substituting into the integral:

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

### Step 3: Solve the Integral Using Partial Fraction Decomposition
The integral \(\int \frac{1}{1 - u^2} \, du\) can be solved by recognizing it as a standard form:

\[
\int \frac{1}{1 - u^2} \, du = \frac{1}{2} \ln \left|\frac{1+u}{1-u}\right| + C
\]

### Step 4: Substitute Back
Substitute \( u = \cos(2x) \) back into the expression:

\[
-\frac{1}{2} \cdot \frac{1}{2} \ln \left|\frac{1+\cos(2x)}{1-\cos(2x)}\right| + C = -\frac{1}{4} \ln \left|\frac{1+\cos(2x)}{1-\cos(2x)}\right| + C
\]

### Step 5: Simplify the Expression
Using the identity \( \frac{1+\cos(2x)}{1-\cos(2x)} = \cot^2(x) \), the integral simplifies to:

\[
-\frac{1}{4} \ln(\cot^2(x)) + C = -\frac{1}{2} \ln|\cot(x)| + C
\]

Thus, the final answer is:

\[
\int \frac{1}{\sin^3(2x)} \, dx = -\frac{1}{2} \ln|\cot(x)| + C
\]

### Summary
The integral \(\int \frac{1}{\sin^3(2x)} \, dx\) evaluates to \(-\frac{1}{2} \ln|\cot(x)| + C\), where \(C\) is the constant of integration.

Would you like a detailed explanation of any specific step? Here are some related problems you might want to try:

1. Solve \(\int \frac{1}{\sin^2(x)} \, dx\).
2. Evaluate \(\int \frac{1}{\sin^4(x)} \, dx\).
3. Solve \(\int \frac{1}{\cos^3(x)} \, dx\).
4. Evaluate \(\int \frac{\sin(x)}{\cos^2(x)} \, dx\).
5. Solve \(\int \frac{1}{\tan(x)} \, dx\).

**Tip:** When dealing with integrals of trigonometric functions, try to simplify using identities and consider substitution methods.

Learn how to solve the integral ∫1/sin(2x)^3 with a detailed step-by-step solution using trigonometric identities, substitution, and partial fraction decomposition. This problem is suitable for advanced high school students and college-level calculus courses.

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