The integral in the image is:

$\int_1^2 \frac{4y^2 - 7y - 12}{y(y+2)(y-3)} \, dy$

We can proceed by simplifying the integrand through partial fraction decomposition. Let me work through the details of this integral for you.

Steps:

1. Perform partial fraction decomposition for the expression $\frac{4y^2 - 7y - 12}{y(y+2)(y-3)}$.
2. Solve for the constants.
3. Integrate each term separately.
4. Evaluate the definite integral from 1 to 2.

I'll start with the partial fraction decomposition.The partial fraction decomposition of the expression is:

$\frac{4y^2 - 7y - 12}{y(y+2)(y-3)} = \frac{2}{y} + \frac{9}{5(y+2)} + \frac{1}{5(y-3)}$

Now, I will integrate each term separately and evaluate the definite integral from $1$ to $2$.The result of the definite integral is:

$\int_1^2 \frac{4y^2 - 7y - 12}{y(y+2)(y-3)} \, dy = \frac{27}{5} \log(2) - \frac{9}{5} \log(3)$

Would you like more details or a breakdown of any specific steps? Here are some related questions for deeper understanding:

1. How is partial fraction decomposition used in integration?
2. What are the properties of logarithms used in solving the integral?
3. How can we check if the decomposition was done correctly?
4. How do you handle improper integrals with logarithmic terms?
5. What are the common strategies for dealing with integrals involving rational functions?

Tip: Partial fraction decomposition is a powerful tool for simplifying rational functions, making them easier to integrate.