I see you have uploaded a photo with a mathematical expression involving an integral. Let's break it down and solve it step by step.

The integral is:

$\int \frac{6x^5 - 2x^4 + 15x^3 + 12x^2 - 2x + 20}{3x^3 - x^2 + 4} \, dx$

Step-by-Step Solution:

1. Check if the numerator is divisible by the denominator:

   • We first look at the degree of the numerator and denominator. The numerator is a polynomial of degree 5, while the denominator is a polynomial of degree 3.

2. Polynomial Long Division: To simplify the integral, we first perform polynomial long division.

   • Divide $6x^5 - 2x^4 + 15x^3 + 12x^2 - 2x + 20$ by $3x^3 - x^2 + 4$.

   Let's begin with the division:

   • The leading term of the numerator is $6x^5$, and the leading term of the denominator is $3x^3$.
   • Divide $6x^5$ by $3x^3$, which gives $2x^2$.
   • Multiply $2x^2$ by the entire denominator, $3x^3 - x^2 + 4$, and subtract the result from the numerator.

   Repeat the process until the degree of the remainder is lower than the denominator's degree.

3. Simplify the result: Once we divide the polynomials, we can break the integral into simpler terms (if possible). If there is a remainder, we need to integrate each term separately.

4. Integrate the result: After simplification, we integrate each term of the quotient and the remainder, if any.

Let me perform the polynomial division to continue the solution for you. sympy as sp

Define the variable and polynomials

x = sp.symbols('x') numerator = 6x**5 - 2x4 + 15*x3 + 12x**2 - 2x + 20 denominator = 3*x3 - x2 + 4

Perform polynomial division

quotient, remainder = sp.div(numerator, denominator, domain='QQ')

quotient, remainderAfter performing the polynomial long division, we get:

$\frac{6x^5 - 2x^4 + 15x^3 + 12x^2 - 2x + 20}{3x^3 - x^2 + 4} = 2x^2 + 5 + \frac{9x^2 - 2x}{3x^3 - x^2 + 4}$

Thus, the integral becomes:

$\int \left(2x^2 + 5 + \frac{9x^2 - 2x}{3x^3 - x^2 + 4} \right) \, dx$

Step 1: Integrate the first part

The first two terms, $2x^2 + 5$, are straightforward to integrate:

$\int (2x^2 + 5) \, dx = \frac{2x^3}{3} + 5x + C_1$

Step 2: Integrate the second part

For the second term $\frac{9x^2 - 2x}{3x^3 - x^2 + 4}$, we use substitution. Let's define:

$u = 3x^3 - x^2 + 4$

Then, we calculate the derivative of $u$:

$\frac{du}{dx} = 9x^2 - 2x$

Thus, the integral becomes:

$\int \frac{9x^2 - 2x}{3x^3 - x^2 + 4} \, dx = \int \frac{du}{u} = \ln |u| + C_2 = \ln |3x^3 - x^2 + 4| + C_2$

Final Answer:

Putting everything together:

$\int \frac{6x^5 - 2x^4 + 15x^3 + 12x^2 - 2x + 20}{3x^3 - x^2 + 4} \, dx = \frac{2x^3}{3} + 5x + \ln |3x^3 - x^2 + 4| + C$

Where $C = C_1 + C_2$ is the constant of integration.

Do you need further details or have any questions on this? Here are some related topics you might find useful:

1. How to perform polynomial long division.
2. The concept of substitution in integration.
3. Properties of logarithmic integrals.
4. Handling higher-degree polynomials in integrals.
5. The role of constant terms in integration.

Tip: Always try to simplify the integrand first before integrating, especially when dealing with polynomials or rational expressions.