To solve the problem, I will analyze the provided image step by step. The question appears to involve simplifying and solving an integral, which includes a square root in the denominator and requires substitution to proceed. Let's focus on translating the task and solving it.

---

Transcription of the task from the image:

We are asked to evaluate the following integral:

$\int \frac{3x - 1}{\sqrt{x^2 - 8x + 32}} \, dx$

Steps provided in the example:

1. Complete the square in the denominator: $x^2 - 8x + 32 = (x - 4)^2 + 16$ So, the integral becomes: $\int \frac{3x - 1}{\sqrt{(x - 4)^2 + 16}} \, dx$

2. Substitute the variable: Let $t = x - 4$, so $x = t + 4$ and $dx = dt$.

   Substituting these values into the integral gives: $\int \frac{3(t + 4) - 1}{\sqrt{t^2 + 16}} \, dt = \int \frac{3t + 11}{\sqrt{t^2 + 16}} \, dt$

3. Split the integral: $\int \frac{3t + 11}{\sqrt{t^2 + 16}} \, dt = \int \frac{3t}{\sqrt{t^2 + 16}} \, dt + \int \frac{11}{\sqrt{t^2 + 16}} \, dt$

   Each term is now integrated separately.

4. Integrate the first term: For $\int \frac{3t}{\sqrt{t^2 + 16}} \, dt$, use substitution $u = t^2 + 16$, so $du = 2t \, dt$.

   The integral becomes: $\int \frac{3t}{\sqrt{t^2 + 16}} \, dt = 3 \int \frac{t}{\sqrt{t^2 + 16}} \, dt = 3 \cdot \sqrt{t^2 + 16}$

5. Integrate the second term: For $\int \frac{11}{\sqrt{t^2 + 16}} \, dt$, recognize this as a standard integral of the form: $\int \frac{1}{\sqrt{t^2 + a^2}} \, dt = \ln|t + \sqrt{t^2 + a^2}|$ Here, $a = 4$, so: $\int \frac{11}{\sqrt{t^2 + 16}} \, dt = 11 \ln|t + \sqrt{t^2 + 16}|$

6. Combine the results: Adding the two parts together, we have: $\int \frac{3t + 11}{\sqrt{t^2 + 16}} \, dt = 3\sqrt{t^2 + 16} + 11\ln|t + \sqrt{t^2 + 16}|$

7. Return to the original variable $x$: Recall that $t = x - 4$, so $t^2 + 16 = (x - 4)^2 + 16$. Substituting back: $\int \frac{3x - 1}{\sqrt{x^2 - 8x + 32}} \, dx = 3\sqrt{(x - 4)^2 + 16} + 11\ln|x - 4 + \sqrt{(x - 4)^2 + 16}| + C$

---

Final Answer:

$\int \frac{3x - 1}{\sqrt{x^2 - 8x + 32}} \, dx = 3\sqrt{(x - 4)^2 + 16} + 11\ln|x - 4 + \sqrt{(x - 4)^2 + 16}| + C$

---

Would you like further clarification on any steps?

Related Questions:

1. How do you complete the square for quadratic expressions?
2. What are standard integrals involving square roots, such as $\sqrt{t^2 + a^2}$?
3. Why is substitution useful in solving integrals like these?
4. Can we solve this integral using another method, such as trigonometric substitution?
5. How do you determine the constant of integration $C$ for definite integrals?

Tip:

When solving integrals involving quadratic expressions, completing the square is often the first critical step to simplify the denominator!