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The given question asks for the result of transforming the integral:

\[
\int_3^8 (5x + 2) \sqrt{x + 1} \, dx
\]

by substituting \( t = \sqrt{x + 1} \).

Let's break down the solution step by step.

### Step 1: Substitution
Given \( t = \sqrt{x + 1} \), we can express \( x \) in terms of \( t \):

\[
t = \sqrt{x + 1} \implies t^2 = x + 1 \implies x = t^2 - 1
\]

### Step 2: Differentiate \( t \)
Now differentiate \( t = \sqrt{x + 1} \) with respect to \( x \):

\[
\frac{dt}{dx} = \frac{1}{2\sqrt{x+1}} = \frac{1}{2t}
\]

Thus, \( dx = 2t \, dt \).

### Step 3: Update the limits of integration
When \( x = 3 \):

\[
t = \sqrt{3 + 1} = \sqrt{4} = 2
\]

When \( x = 8 \):

\[
t = \sqrt{8 + 1} = \sqrt{9} = 3
\]

So the new limits for \( t \) are from \( t = 2 \) to \( t = 3 \).

### Step 4: Update the integral
Substitute \( x = t^2 - 1 \) and \( dx = 2t \, dt \) into the original integral:

\[
\int_3^8 (5x + 2) \sqrt{x + 1} \, dx = \int_2^3 (5(t^2 - 1) + 2) t \cdot 2t \, dt
\]

Simplify the integrand:

\[
(5(t^2 - 1) + 2) = 5t^2 - 5 + 2 = 5t^2 - 3
\]

Thus, the integral becomes:

\[
2 \int_2^3 (5t^2 - 3) t^2 \, dt
\]

### Final Answer:
The transformed integral is:

\[
2 \int_2^3 (5t^2 - 3) t^2 \, dt
\]

This corresponds to the second option in the provided choices.

### Would you like more details or have any other questions?

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Here are 5 related questions for practice:
1. What is the general method for substitution in definite integrals?
2. How does the chain rule apply when differentiating a substitution?
3. How can substitution simplify integrals involving square roots?
4. How do you update limits of integration after a substitution?
5. What are some common substitutions for integrals involving square roots?

**Tip:** Always remember to adjust the limits of integration when performing a substitution in a definite integral.

После подстановки t = √(x + 1) интеграл ∫(5x + 2)√(x + 1) dx от 3 до 8 примет вид:

This detailed guide walks you through the process of solving the integral ∫(5x + 2)√(x + 1) dx from 3 to 8 using the substitution t = √(x + 1). Learn the key steps of substitution, updating limits of integration, and simplifying the integral expression. Suitable for undergraduate-level calculus students.

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