https://file.aiquickdraw.com/mathbot/file/17253827484061ftbr0wp.jpg

The integral provided in the image is:

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

To evaluate this integral, we can use a substitution. Let's set:

\[
u = \sin x \implies du = \cos x \, dx
\]

Substituting these into the integral, we get:

\[
\int \frac{\cos x \, dx}{\sin x} = \int \frac{du}{u}
\]

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln |u|\). So, the solution becomes:

\[
\int \frac{du}{u} = \ln |u| + C
\]

Substituting back for \(u = \sin x\), we have:

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

where \(C\) is the constant of integration.

### Summary
The integral \(\int \frac{\cos x \, dx}{\sin x}\) evaluates to \(\ln |\sin x| + C\).

Would you like to see more details on the steps or have any questions?

### Follow-up Questions:
1. What is the general approach for evaluating integrals involving trigonometric functions?
2. Can this integral be solved using another method, such as integration by parts?
3. How would the integral change if it were \(\int \frac{\sin x \, dx}{\cos x}\)?
4. What are some common substitutions used in trigonometric integrals?
5. How can we verify the result of this integral by differentiation?

### Tip:
When working with integrals involving trigonometric functions, looking for a substitution that simplifies the integrand can often make the problem much easier to solve.

Evaluate the integral ∫ (cos x dx) / (sin x)

Learn how to evaluate the integral ∫ (cos x dx) / (sin x) using substitution in this detailed solution. This problem covers key concepts in calculus and trigonometric functions, making it suitable for students in Grades 11-12.

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