$$ \int_{}^{}\!\frac{\sin\left(x\right)}{\text{cos(x)}}\,\mathrm{d}x $$

The given integral is:

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

This can be simplified by recognizing that $\frac{\sin(x)}{\cos(x)} = \tan(x)$. So the integral becomes:

\[
\int \tan(x) \, dx
\]

The integral of $\tan(x)$ is a standard result:

\[
\int \tan(x) \, dx = \ln|\sec(x)| + C
\]

Thus, the solution is:

\[
\int \frac{\sin(x)}{\cos(x)} \, dx = \ln|\sec(x)| + C
\]

where $C$ is the constant of integration.

Would you like more details or further explanation?

Here are 5 related questions to explore:
1. What is the integral of $\sec(x)$?
2. How do you integrate $\tan^2(x)$?
3. Can you differentiate $\ln|\sec(x)|$ to check the result?
4. What is the integral of $\frac{\sin^2(x)}{\cos^2(x)}$?
5. How would you solve $\int \tan^3(x) \, dx$?

**Tip:** When integrating trigonometric functions, simplifying the expression with identities can often make the process easier.

Learn how to compute the integral of sin(x)/cos(x), which simplifies to the standard result for ∫tan(x) dx = ln|sec(x)| + C. This solution involves basic trigonometric identities and integration techniques, suitable for students in Grades 11-12.

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