To solve the integral of \(\frac{7 \sin(x) + 4 \sec(x)}{\tan(x)} \, dx\), let's break this down step by step.

First, express \(\tan(x)\), \(\sec(x)\), and \(\sin(x)\) in terms of basic trigonometric identities:

\[
\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \sec(x) = \frac{1}{\cos(x)}
\]

Now, the expression becomes:

\[
\frac{7 \sin(x) + 4 \sec(x)}{\tan(x)} = \frac{7 \sin(x) + \frac{4}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}}
\]

Simplify this expression:

\[
= \frac{(7 \sin(x) + \frac{4}{\cos(x)}) \cos(x)}{\sin(x)}
\]

Distribute \(\cos(x)\) to both terms in the numerator:

\[
= \frac{7 \sin(x) \cos(x) + 4}{\sin(x)}
\]

Now, separate the terms in the numerator:

\[
= \frac{7 \sin(x) \cos(x)}{\sin(x)} + \frac{4}{\sin(x)}
\]

Simplify each term:

\[
= 7 \cos(x) + 4 \csc(x)
\]

Now, the integral becomes:

\[
\int \left(7 \cos(x) + 4 \csc(x)\right) dx
\]

We can split this integral into two parts:

\[
\int 7 \cos(x) \, dx + \int 4 \csc(x) \, dx
\]

### First Integral:
\[
\int 7 \cos(x) \, dx = 7 \sin(x)
\]

### Second Integral:
\[
\int 4 \csc(x) \, dx = 4 \ln|\csc(x) - \cot(x)|
\]

Thus, the solution to the integral is:

\[
7 \sin(x) + 4 \ln|\csc(x) - \cot(x)| + C
\]

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

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### Related Questions:
1. What is the integral of \( \sec(x) \)?
2. How can we derive the integral of \( \csc(x) \)?
3. What are the key trigonometric identities involved in integration?
4. How can you simplify complex trigonometric integrals?
5. How do you handle integrals with products of trigonometric functions?

### Tip:
When solving integrals involving trigonometric functions, it is helpful to recall the basic identities, especially those involving \( \sec(x) \), \( \csc(x) \), and \( \tan(x) \).

This page provides a detailed step-by-step solution for the integral of (7 sin(x) + 4 sec(x)) / tan(x). It covers key trigonometric identities and techniques for simplifying and solving complex integrals involving sine, secant, and tangent functions. Suitable for college-level calculus students.

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