The first problem in your image asks to evaluate the integral $\int 5 \ln(x^2 - x + 8) \, dx$, using integration by parts and techniques discussed in that section. Let's go step by step:

Problem:

Evaluate the integral: $\int 5 \ln(x^2 - x + 8) \, dx$

Solution:

1. Choose substitution for integration by parts: We apply integration by parts where: $u = \ln(x^2 - x + 8) \quad \text{and} \quad dv = 5 \, dx$ Now, compute the differential terms:

   \quad \text{and} \quad v = 5x$$

2. Apply the integration by parts formula: Integration by parts is given by: $\int u \, dv = uv - \int v \, du$ Using this, we have: $\int 5 \ln(x^2 - x + 8) \, dx = 5x \ln(x^2 - x + 8) - \int 5x \cdot \frac{2x - 1}{x^2 - x + 8} \, dx$

3. Simplify the remaining integral: The remaining integral involves simplifying the expression: $\int \frac{5x(2x - 1)}{x^2 - x + 8} \, dx$ At this point, you can use further substitution or decomposition methods to simplify and solve the remaining integral.

This outlines the general method to solve the integral using integration by parts.

For the second question in the image:

Problem:

Evaluate the integral: $\int \frac{ax}{x^2 - bx} \, dx$

Solution:

1. Simplify the denominator: Factor the denominator: $x^2 - bx = x(x - b)$

2. Substitute: Let: $u = x^2 - bx \quad \Rightarrow \quad du = (2x - b) \, dx$ Solve the integral using this substitution, simplifying and solving the resulting expression.

Would you like more details on solving the integrals or help with any specific step? Let me know!

Related Questions:

1. How do you perform substitution in integrals with logarithmic functions?
2. What are the typical steps for integration by parts?
3. Can you explain the method of partial fraction decomposition in solving integrals?
4. How do you handle integrals involving rational functions like $\frac{x}{x^2 - ax + b}$?
5. What are common integral techniques involving logarithmic and polynomial expressions?

Tip:

When dealing with integrals of logarithmic functions, remember that integration by parts is often a helpful approach, particularly if you can pair the logarithmic function with a polynomial term.