The image contains two integral problems. Let's break them down and solve them step by step.

1. First Problem:

$\int (6x^2 + 8x - 5) \, dx$

This is an indefinite integral of the polynomial function $6x^2 + 8x - 5$. We solve it by integrating each term separately.

Solution:

1. $\int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} = 2x^3$
2. $\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2$
3. $\int -5 \, dx = -5x$

Thus, the integral is: $2x^3 + 4x^2 - 5x + C$ where $C$ is the constant of integration.

2. Second Problem:

$\int_1^2 (9x^2 - 4x^3 + 2x) \, dx$

This is a definite integral over the interval [1, 2]. We solve it by integrating each term separately and then evaluating the result from 1 to 2.

Solution:

1. $\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3$
2. $\int -4x^3 \, dx = -4 \cdot \frac{x^4}{4} = -x^4$
3. $\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2$

Now, we evaluate the resulting function $3x^3 - x^4 + x^2$ at the limits 2 and 1: [ \left[ 3(2)^3 - (2)^4 + (2)^2 \right] - \left[ 3(1)^3 - (1)^4 + (1)^2 \right] ] Simplifying: [ \left[ 3(8) - 16 + 4 \right] - \left[ 3(1) - 1 + 1 \right] ] [ \left[ 24 - 16 + 4 \right] - \left[ 3 - 1 + 1 \right] ] [ \left[ 12 \right] - \left[ 3 \right] = 9 ]

Thus, the value of the definite integral is $9$.

Would you like more details or have any questions?
Here are 5 related questions that expand on this topic:

1. How do you solve integrals involving trigonometric functions?
2. What are some common methods for evaluating definite integrals?
3. Can integrals be used to find the area between curves?
4. How does substitution work in integrals, and when should it be used?
5. What is the importance of the constant $C$ in indefinite integrals?

Tip: When solving definite integrals, always carefully check the limits of integration and simplify the result step by step to avoid errors.