To solve the integral:

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

### Step 1: Simplify \(\sin(2x)\)

Using the identity for \(\sin(2x)\), we can rewrite it as:

\[
\sin(2x) = 2\sin(x)\cos(x)
\]

Thus, the integral becomes:

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

### Step 2: Substitution
We can apply the substitution method to make this integral easier. Let:

\[
u = \cos(x)
\]
Then, the derivative of \(u\) with respect to \(x\) is:

\[
du = -\sin(x) \, dx
\]

Therefore, the integral becomes:

\[
-2 \int \frac{u}{1 + u^2} \, du
\]

### Step 3: Solve the Integral
The integral of \(\frac{u}{1 + u^2}\) is a straightforward one, which can be done using a simple substitution or recognizing it as a standard form. We know:

\[
\int \frac{u}{1 + u^2} \, du = \frac{1}{2} \ln(1 + u^2)
\]

Thus, the integral becomes:

\[
-2 \times \frac{1}{2} \ln(1 + u^2) + C = -\ln(1 + u^2) + C
\]

### Step 4: Substitute Back \(u = \cos(x)\)

Finally, substitute back \(u = \cos(x)\):

\[
-\ln(1 + \cos^2(x)) + C
\]

### Final Answer:

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

Would you like a deeper breakdown or clarification on any step?

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Here are 5 related questions to extend your understanding:

1. How can substitution be applied to other trigonometric integrals?
2. What are common techniques to simplify integrals with trigonometric functions?
3. Can the integral of \(\frac{1}{1 + \cos^2(x)}\) be solved without substitution?
4. What other identities can simplify integrals involving \(\sin(2x)\)?
5. How does the integral of \(\sin^2(x)\) differ from this one?

**Tip**: Memorizing standard trigonometric integrals can significantly speed up problem-solving!

Learn how to solve the trigonometric integral of sin(2x)/(1 + cos^2(x)) with detailed steps, including substitution and logarithmic integration. This problem involves key calculus concepts and is ideal for students studying undergraduate-level calculus.

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