https://file.aiquickdraw.com/mathbot/file/17263933470575ievm8fb.jpg

The integral in the image appears to be:

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

This integral can be simplified by using a trigonometric identity. Recall that:

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

Thus, we can rewrite the integrand as:

\[
\frac{1}{\sin(x) \cos(x)} = \frac{2}{\sin(2x)}
\]

This simplifies the integral to:

\[
\int \frac{dx}{\sin(x) \cos(x)} = 2 \int \frac{dx}{\sin(2x)}
\]

The integral of \( \frac{1}{\sin(2x)} \) is known, and it is the logarithmic form:

\[
\int \frac{dx}{\sin(2x)} = -\frac{1}{2} \ln|\cot(2x)|
\]

Thus, the final result of the original integral is:

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

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

Would you like more details on any part of the solution?

Here are some related questions for further exploration:
1. How can other trigonometric identities help simplify similar integrals?
2. What is the integral of \( \frac{1}{\cos^2(x)} \)?
3. How can substitution methods simplify trigonometric integrals?
4. Can the integral of \( \frac{1}{\sin^2(x)} \) be calculated similarly?
5. What is the relationship between logarithmic functions and trigonometric integrals?

**Tip:** Always look for ways to simplify trigonometric expressions using identities like double-angle formulas!

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

Learn how to solve the integral \( \int \frac{dx}{\sin(x) \cos(x)} \) by applying the double-angle formula for sine. This detailed solution simplifies the expression using trigonometric identities, resulting in a logarithmic solution. Suitable for advanced high school or early university students.

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