https://file.aiquickdraw.com/mathbot/file/1736094277929qi0ze7ac.jpg

The integral shown in the image is:

\[
\int \frac{\sin(x)}{\cos(x)} \, dx
\]

### Solution:
This integral can be solved using substitution.

1. **Rewrite the integrand:**
   \[
   \frac{\sin(x)}{\cos(x)} = \tan(x)
   \]

   So the integral becomes:
   \[
   \int \tan(x) \, dx
   \]

2. **Standard formula:**
   The integral of \(\tan(x)\) is a standard result:
   \[
   \int \tan(x) \, dx = -\ln|\cos(x)| + C
   \]

3. **Final answer:**
   \[
   \int \frac{\sin(x)}{\cos(x)} \, dx = -\ln|\cos(x)| + C
   \]

### Explanation:
This result comes from recognizing that the derivative of \(\cos(x)\) is \(-\sin(x)\), and using the substitution \(u = \cos(x)\), leading directly to the logarithmic result.

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Would you like more details or an explanation of how the substitution works?

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### Related Questions:
1. How would this integral change if the numerator was \(\cos(x)\) instead of \(\sin(x)\)?
2. What is the integral of \(\tan^2(x)\)?
3. How do we derive the standard formula for \(\int \tan(x) \, dx\)?
4. How does this result relate to the integral of \(\cot(x)\)?
5. What happens to the integral if the limits are added, for example, from \(0\) to \(\pi/4\)?

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### Tip:
When solving trigonometric integrals, recognizing standard identities or rewriting the function in terms of \(\sin(x)\) and \(\cos(x)\) is often the key to simplifying the problem.

Evaluate the integral \( \int \frac{\sin(x)}{\cos(x)} dx \).

Learn how to evaluate the integral of \( \frac{\sin(x)}{\cos(x)} \) using substitution and standard formulas. This step-by-step solution covers trigonometric identities and integral calculus concepts, suitable for high school and college-level students.

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