The integral given in the image is:

$\int \frac{x^3 - 4x^2 - 4x + 20}{x^2 - 5} \, dx$

To solve this, the first step would typically involve dividing the numerator by the denominator using polynomial long division, as the degree of the numerator is greater than or equal to the degree of the denominator.

Let me perform the polynomial division and proceed with the solution.The polynomial division of $\frac{x^3 - 4x^2 - 4x + 20}{x^2 - 5}$ gives the quotient $x - 4$ and the remainder $x$.

Thus, we can rewrite the integral as:

$\int \frac{x^3 - 4x^2 - 4x + 20}{x^2 - 5} \, dx = \int (x - 4) \, dx + \int \frac{x}{x^2 - 5} \, dx$

Now, let's solve these integrals individually:

1. The first integral $\int (x - 4) \, dx$ is straightforward.
2. The second integral $\int \frac{x}{x^2 - 5} \, dx$ can be solved by a substitution.

Let me compute both integrals.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can proceed with the following steps to complete the problem:

1. First integral: $\int (x - 4) \, dx = \frac{x^2}{2} - 4x + C_1$.

2. Second integral: $\int \frac{x}{x^2 - 5} \, dx$ can be solved by substitution. Let $u = x^2 - 5$, so that $du = 2x \, dx$, and this integral becomes $\frac{1}{2} \ln |x^2 - 5| + C_2$.

Thus, the solution for the whole integral would be:

$\frac{x^2}{2} - 4x + \frac{1}{2} \ln |x^2 - 5| + C$

If you have any more questions or would like further explanation, let me know!

Related Questions:

1. How do you perform polynomial division for rational functions?
2. What is the method of substitution in integrals, and when is it useful?
3. How does the natural logarithm arise in integral solutions like this one?
4. What are other techniques for integrating rational functions?
5. How does the constant of integration $C$ play a role in indefinite integrals?

Tip:

When you face an integral involving a rational function, always check if polynomial division is possible to simplify the expression.