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To solve the integral:

\[
\int \frac{e^x + \sin x + 4\cos x + 3}{e^x + 5\sin x + 3\cos x + 9} \, dx
\]

we will first analyze the structure of the integrand. The numerator and denominator are both functions of \(x\), involving exponentials, sine, and cosine terms. The goal will be to find a simpler way to integrate this expression, potentially by simplifying or applying substitution techniques.

### Step 1: Analyze the structure

The given integral can be viewed as a rational function, and simplification by substitution could be a good strategy. Let's explore this by trying a substitution based on the exponential function \( e^x \).

Let \( u = e^x + 5\sin x + 3\cos x + 9 \), which is the denominator.

### Step 2: Differentiate \( u \)

We differentiate \( u \) with respect to \( x \):

\[
du = \left(e^x + 5\cos x - 3\sin x\right) dx
\]

### Step 3: Compare the numerator

Now compare this differential with the numerator. The numerator is:

\[
e^x + \sin x + 4\cos x + 3
\]

We notice that the terms are similar to those in \( du \), except for a few constants and coefficients. This suggests that the substitution method will simplify the integral substantially.

### Step 4: Express the integral in terms of \( u \)

Using the substitution \( u = e^x + 5\sin x + 3\cos x + 9 \), we can rewrite the integral as:

\[
\int \frac{du}{u}
\]

### Step 5: Integrate

The integral of \( \frac{1}{u} \) is:

\[
\ln |u| + C
\]

Substituting back \( u = e^x + 5\sin x + 3\cos x + 9 \), we get:

\[
\ln |e^x + 5\sin x + 3\cos x + 9| + C
\]

Thus, the solution to the integral is:

\[
\boxed{\ln |e^x + 5\sin x + 3\cos x + 9| + C}
\]

Would you like any additional details or explanations on any steps? 

Here are some related questions you might find interesting:

1. How would this solution change if the denominator was \( e^x + \sin x + \cos x + 1 \)?
2. Can integration by parts be applied to this type of integral?
3. What are some other substitution techniques that can be used for complex integrals?
4. How does the structure of this integral compare to simpler rational functions?
5. What methods are used to solve integrals involving more complicated trigonometric identities?

**Tip:** Substitution is one of the most effective techniques when dealing with integrals that involve functions whose derivatives appear in the integrand.

Solve the integral: ∫ (e^x + sin x + 4cos x + 3) / (e^x + 5sin x + 3cos x + 9) dx

Learn how to solve the integral ∫ (e^x + sin x + 4cos x + 3) / (e^x + 5sin x + 3cos x + 9) dx using substitution. This problem explores the substitution method and integration of rational functions, involving both trigonometric and exponential terms.

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