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The problem provided is to express the integrand as a sum of partial fractions and then evaluate the integral:

\[
\int \frac{4x + 13}{x^2 + 9x + 14} \, dx
\]

### Step 1: Factor the denominator
The quadratic expression \( x^2 + 9x + 14 \) needs to be factored. To factor it, we find two numbers that multiply to 14 and add to 9. These numbers are 7 and 2.

So,
\[
x^2 + 9x + 14 = (x + 7)(x + 2)
\]

### Step 2: Set up partial fraction decomposition
We want to express the integrand as a sum of partial fractions. This means we write:
\[
\frac{4x + 13}{(x + 7)(x + 2)} = \frac{A}{x + 7} + \frac{B}{x + 2}
\]

Multiply both sides by the denominator \( (x + 7)(x + 2) \) to eliminate the denominators:
\[
4x + 13 = A(x + 2) + B(x + 7)
\]

### Step 3: Solve for \(A\) and \(B\)
Expand both sides:
\[
4x + 13 = A(x + 2) + B(x + 7) = A x + 2A + B x + 7B
\]
\[
4x + 13 = (A + B)x + (2A + 7B)
\]

Now, equate the coefficients of \(x\) and the constant terms:
- Coefficient of \(x\): \( A + B = 4 \)
- Constant term: \( 2A + 7B = 13 \)

Solve this system of equations:
1. \( A + B = 4 \)
2. \( 2A + 7B = 13 \)

From equation (1): \( A = 4 - B \). Substitute this into equation (2):
\[
2(4 - B) + 7B = 13
\]
\[
8 - 2B + 7B = 13
\]
\[
8 + 5B = 13
\]
\[
5B = 5 \quad \Rightarrow \quad B = 1
\]

Now substitute \( B = 1 \) into equation (1):
\[
A + 1 = 4 \quad \Rightarrow \quad A = 3
\]

### Step 4: Rewrite the integrand
Now we have:
\[
\frac{4x + 13}{(x + 7)(x + 2)} = \frac{3}{x + 7} + \frac{1}{x + 2}
\]

### Step 5: Integrate each term
Now we integrate term by term:
\[
\int \frac{3}{x + 7} \, dx + \int \frac{1}{x + 2} \, dx
\]

The integrals are straightforward:
\[
\int \frac{3}{x + 7} \, dx = 3 \ln |x + 7| + C_1
\]
\[
\int \frac{1}{x + 2} \, dx = \ln |x + 2| + C_2
\]

### Step 6: Final answer
Combining the two integrals and constants:
\[
\int \frac{4x + 13}{x^2 + 9x + 14} \, dx = 3 \ln |x + 7| + \ln |x + 2| + C
\]
where \( C = C_1 + C_2 \) is the constant of integration.

Would you like further details on any step or more explanation?

Here are 5 related questions:
1. How do you solve a system of linear equations like the one used to find \( A \) and \( B \)?
2. What are the different techniques for factoring quadratic expressions?
3. How do you apply the method of partial fractions to higher-degree polynomials?
4. What are the common properties of logarithmic integrals?
5. How can you verify the result of an integral through differentiation?

**Tip:** For integrals involving quadratic denominators that don't factor easily, consider completing the square or using trigonometric substitutions.

Express the integrand as a sum of partial fractions and evaluate the integral: ∫ (4x + 13) / (x^2 + 9x + 14) dx.

Learn how to express the integrand ∫ (4x + 13) / (x^2 + 9x + 14) dx as a sum of partial fractions and evaluate the integral. This step-by-step solution includes factoring the quadratic denominator, solving for constants, and integrating logarithmic functions.

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