We are tasked with finding the indefinite integral of the function:

$\int \frac{x^2}{x - 4} \, dx$

To solve this, we will use polynomial long division because the degree of the numerator $x^2$ is higher than the degree of the denominator $x - 4$.

Step 1: Perform polynomial long division

We will divide $x^2$ by $x - 4$.

1. Divide the leading term $x^2$ by the leading term $x$, which gives $x$.
2. Multiply $x$ by $x - 4$, yielding $x(x - 4) = x^2 - 4x$.
3. Subtract $x^2 - 4x$ from $x^2$, which leaves a remainder of $4x$.

Now, we repeat the process for the remainder $4x$:

4. Divide $4x$ by $x$, which gives $+4$.
5. Multiply $4$ by $x - 4$, yielding $4(x - 4) = 4x - 16$.
6. Subtract $4x - 16$ from $4x$, which leaves a remainder of $+16$.

Thus, the division gives:

$\frac{x^2}{x - 4} = x + 4 + \frac{16}{x - 4}$

Step 2: Integrate each term

Now that we've simplified the integrand, we can integrate term by term:

$\int \left( x + 4 + \frac{16}{x - 4} \right) \, dx$

• The integral of $x$ is $\frac{x^2}{2}$.
• The integral of $4$ is $4x$.
• The integral of $\frac{16}{x - 4}$ is $16 \ln |x - 4|$ (we use absolute value because the logarithm requires it).

Step 3: Write the final answer

Combining all the terms, we get the indefinite integral:

$\frac{x^2}{2} + 4x + 16 \ln |x - 4| + C$

where $C$ is the constant of integration.

Would you like further details on this solution, or have any questions?

Related Questions:

1. How do you perform polynomial long division in general cases?
2. What is the role of absolute values in logarithmic integrals?
3. How can we handle improper integrals involving division by polynomials?
4. How do we integrate rational functions of higher degrees?
5. What techniques can be used to check the correctness of an indefinite integral?

Tip:

Always simplify the integrand as much as possible before attempting integration; this can often lead to easier calculations.