https://file.aiquickdraw.com/mathbot/file/1728298618945cvsgockq.jpg

Let's analyze the integral problem from the image:

\[
\int_0^{\frac{\pi}{4}} \frac{\ln(\tan(x) \cdot \tan(2x))}{\cos^2(x)} \, dx = -\ln(2)
\]

### Steps for Solving:

1. **Break down the logarithmic term**:
   \[
   \ln(\tan(x) \cdot \tan(2x)) = \ln(\tan(x)) + \ln(\tan(2x))
   \]
   The integral then becomes:
   \[
   \int_0^{\frac{\pi}{4}} \frac{\ln(\tan(x)) + \ln(\tan(2x))}{\cos^2(x)} \, dx
   \]
   
2. **Split the integral** into two parts:
   \[
   I = \int_0^{\frac{\pi}{4}} \frac{\ln(\tan(x))}{\cos^2(x)} \, dx + \int_0^{\frac{\pi}{4}} \frac{\ln(\tan(2x))}{\cos^2(x)} \, dx
   \]

3. **Simplify the second term** using substitution. Set \( u = 2x \), then \( du = 2dx \). The bounds for this new integral will change from \( 0 \) to \( \frac{\pi}{4} \) to \( 0 \) to \( \frac{\pi}{2} \). This simplifies the second integral significantly.

4. **Combine and solve the integrals**. The result will be a negative logarithmic value, which is given in the problem as \( -\ln(2) \).

This integral is non-trivial and involves advanced techniques such as substitution and exploiting known results from integral tables or symmetry.

Would you like a more detailed breakdown of any of these steps?

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Here are 5 related questions for practice:

1. How can you simplify integrals involving logarithms of trigonometric functions?
2. What substitution could simplify an integral involving \( \tan(2x) \)?
3. How does splitting integrals help in solving complex integrals like this one?
4. What techniques are commonly used to solve integrals over \( \left[ 0, \frac{\pi}{4} \right] \) with trigonometric functions?
5. Why is the final result expressed as \( -\ln(2) \) in this problem?

**Tip**: When dealing with integrals of trigonometric functions, always look for possible substitutions or trigonometric identities that could simplify the expression before integrating.

What is the integral of ln(tan x * tan 2x) / cos^2(x) from 0 to π/4?

Step-by-step solution of the integral problem involving logarithmic and trigonometric functions: ln(tan x * tan 2x) / cos^2(x), using advanced calculus techniques like substitution. Suitable for calculus students.

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