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The image contains two integrals and a substitution hint. You are asked to solve these integrals using the substitution \( \tan\left(\frac{x}{2}\right) = t \). The given integrals are:

1. \( \int \frac{1}{3 + 5 \cos x} \, dx \)
2. \( \int \frac{1}{4 \sin x + 3 \cos x} \, dx \)

Let me guide you through solving these integrals using the Weierstrass substitution \( \tan\left(\frac{x}{2}\right) = t \).

### Step 1: Substitution Relations
Using \( t = \tan\left(\frac{x}{2}\right) \), the following standard relations apply:
- \( dx = \frac{2 \, dt}{1 + t^2} \)
- \( \sin x = \frac{2t}{1 + t^2} \)
- \( \cos x = \frac{1 - t^2}{1 + t^2} \)

### 1. \( \int \frac{1}{3 + 5 \cos x} \, dx \)

Substitute \( \cos x \) and \( dx \) into the integral:
\[
\int \frac{1}{3 + 5 \left(\frac{1 - t^2}{1 + t^2}\right)} \cdot \frac{2 \, dt}{1 + t^2}
\]

Simplifying the expression:
\[
= \int \frac{2}{(3 + 5) + (3 - 5)t^2} \, dt
\]
\[
= \int \frac{2}{8 - 2t^2} \, dt
\]
\[
= \int \frac{1}{4 - t^2} \, dt
\]

This is a standard integral of the form \( \frac{1}{a^2 - t^2} \), whose result is:
\[
= \frac{1}{2} \ln\left|\frac{2 + t}{2 - t}\right| + C
\]

Finally, back-substitute \( t = \tan\left(\frac{x}{2}\right) \) to get the solution.

### 2. \( \int \frac{1}{4 \sin x + 3 \cos x} \, dx \)

Substitute \( \sin x \), \( \cos x \), and \( dx \):
\[
\int \frac{1}{4 \left(\frac{2t}{1 + t^2}\right) + 3 \left(\frac{1 - t^2}{1 + t^2}\right)} \cdot \frac{2 \, dt}{1 + t^2}
\]

Simplifying the expression:
\[
= \int \frac{2}{\frac{8t + 3 - 3t^2}{1 + t^2}} \cdot \frac{1}{1 + t^2} \, dt
\]
\[
= \int \frac{2 (1 + t^2)}{8t + 3 - 3t^2} \, dt
\]

This integral is more complex, but we can attempt partial fraction decomposition or further manipulation. However, this process will involve solving a rational function, which can be done step-by-step depending on the complexity of the resulting expression.

Would you like to go deeper into the second integral, or need clarification on any part of the first solution?

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Here are some follow-up questions you might find useful:
1. Can you express the substitution formula in terms of hyperbolic functions?
2. How do you handle cases where the integrand includes both \( \sin x \) and \( \cos x \)?
3. What other methods could solve these types of trigonometric integrals?
4. How does the Weierstrass substitution simplify integration problems?
5. Can you explain the role of partial fraction decomposition in handling rational functions?

**Tip:** For integrals involving trigonometric functions, Weierstrass substitution often converts the problem into a rational function, simplifying the integration process.

Using the substitution \( \tan\left(\frac{x}{2}\right) = t \), find the following integrals: (1) \( \int \frac{1}{3 + 5 \cos x} \, dx \) (2) \( \int \frac{1}{4 \sin x + 3 \cos x} \, dx \).

Solve integrals using Weierstrass substitution. Find detailed solutions to \( \int \frac{1}{3 + 5 \cos x} \, dx \) and \( \int \frac{1}{4 \sin x + 3 \cos x} \, dx \) using the substitution \( \tan\left(\frac{x}{2}\right) = t \), simplifying trigonometric integrals.

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