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The problem you have shared is:

\[
\int \frac{x^3 \, dx}{x^2 + 7x + 12}
\]

Let's solve this step by step.

### Step 1: Factor the denominator

The denominator is \(x^2 + 7x + 12\). We can factor this quadratic expression:
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

So, the integral becomes:

\[
\int \frac{x^3 \, dx}{(x + 3)(x + 4)}
\]

### Step 2: Polynomial long division

Since the degree of the numerator (3) is higher than that of the denominator (2), we perform polynomial long division. Dividing \(x^3\) by \(x^2 + 7x + 12\) gives:

1. First term: \(\frac{x^3}{x^2} = x\)
2. Multiply \(x(x^2 + 7x + 12) = x^3 + 7x^2 + 12x\)
3. Subtract: \((x^3 - (x^3 + 7x^2 + 12x)) = -7x^2 - 12x\)
4. Second term: \(\frac{-7x^2}{x^2} = -7\)
5. Multiply: \(-7(x^2 + 7x + 12) = -7x^2 - 49x - 84\)
6. Subtract: \((-7x^2 - 12x) - (-7x^2 - 49x - 84) = 37x + 84\)

Now, the division gives:
\[
\frac{x^3}{x^2 + 7x + 12} = x - 7 + \frac{37x + 84}{x^2 + 7x + 12}
\]

### Step 3: Break the integral

Now, we split the integral:

\[
\int \frac{x^3 \, dx}{(x+3)(x+4)} = \int (x - 7) \, dx + \int \frac{37x + 84}{(x+3)(x+4)} \, dx
\]

We can now integrate the first part and proceed to simplify the second part using partial fractions.

### Step 4: Partial fractions

We decompose \(\frac{37x + 84}{(x+3)(x+4)}\) into partial fractions:

\[
\frac{37x + 84}{(x+3)(x+4)} = \frac{A}{x+3} + \frac{B}{x+4}
\]

Multiplying both sides by \((x + 3)(x + 4)\) and equating coefficients:

\[
37x + 84 = A(x + 4) + B(x + 3)
\]
Expanding:
\[
37x + 84 = A(x) + 4A + B(x) + 3B = (A + B)x + (4A + 3B)
\]

Now, equate the coefficients of \(x\) and the constant terms:

1. \(A + B = 37\)
2. \(4A + 3B = 84\)

Solving this system:

From \(A + B = 37\), we have \(B = 37 - A\). Substituting into the second equation:

\[
4A + 3(37 - A) = 84
\]
\[
4A + 111 - 3A = 84
\]
\[
A = 84 - 111 = -27
\]
Substitute \(A = -27\) into \(A + B = 37\):

\[
-27 + B = 37 \quad \Rightarrow \quad B = 64
\]

Thus, the partial fraction decomposition is:

\[
\frac{37x + 84}{(x+3)(x+4)} = \frac{-27}{x+3} + \frac{64}{x+4}
\]

### Step 5: Complete the integral

Now, we can rewrite the original integral as:

\[
\int \frac{x^3 \, dx}{(x+3)(x+4)} = \int (x - 7) \, dx + \int \frac{-27 \, dx}{x+3} + \int \frac{64 \, dx}{x+4}
\]

Integrating each term:

\[
= \frac{x^2}{2} - 7x - 27 \ln|x+3| + 64 \ln|x+4| + C
\]

### Final Answer:

\[
\frac{x^2}{2} - 7x - 27 \ln|x+3| + 64 \ln|x+4| + C
\]

Would you like any further details or explanation?

Here are 5 related questions for further exploration:
1. How does polynomial long division work with higher degree polynomials?
2. What is the general method for partial fraction decomposition?
3. Can you apply the same method if the numerator had a different degree?
4. How would this change if the quadratic in the

Solve the integral of (x^3) / (x^2 + 7x + 12) using polynomial long division and partial fraction decomposition. This step-by-step guide explores the factorization of quadratic expressions, partial fraction techniques, and integration methods. Suitable for undergraduate calculus students.

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