The integral you're looking to solve is:

\[
\int \sec(5x) \tan(5x) \, dx
\]

This is a standard integral. The key idea here is to recognize that the derivative of \(\sec(5x)\) is \(5 \sec(5x) \tan(5x)\). To make use of this, we can apply substitution.

### Step-by-step solution:

1. **Substitute** \( u = 5x \), hence \( du = 5 dx \), or equivalently, \( dx = \frac{du}{5} \).

2. Rewriting the integral in terms of \( u \):
\[
\int \sec(5x) \tan(5x) \, dx = \frac{1}{5} \int \sec(u) \tan(u) \, du
\]

3. Now, use the fact that \( \frac{d}{du} \sec(u) = \sec(u) \tan(u) \). So the integral simplifies to:
\[
\frac{1}{5} \sec(u) + C
\]

4. Substitute back \( u = 5x \):
\[
\frac{1}{5} \sec(5x) + C
\]

Thus, the solution to the integral is:
\[
\boxed{\frac{1}{5} \sec(5x) + C}
\]

Would you like further details on any step? 

Here are 5 related questions to deepen your understanding:
1. How do we know when to use substitution in integrals?
2. What is the derivative of \(\sec(x)\), and how does it relate to this problem?
3. Can this method be used for similar integrals involving trigonometric functions?
4. How would this integral change if it were \(\sec^2(5x)\) instead of \(\sec(5x) \tan(5x)\)?
5. How do you reverse an integral to check if the result is correct?

**Tip:** Always double-check your substitution by differentiating your result to see if it matches the original integrand.

Solve the integration of sec(5x) tan(5x) dx

Learn how to solve the integral ∫ sec(5x) tan(5x) dx using the substitution method. This problem explores key calculus concepts involving trigonometric integrals and provides a detailed step-by-step solution suitable for advanced high school or college-level students.

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