The image you've uploaded illustrates the step-by-step solution for a definite integral problem involving substitution. Here's a detailed explanation of the steps shown:

1. Original Integral: $\int \sqrt{4x + 1} \, dx$

2. Choosing the Substitution: We define a new variable $u$ to simplify the integrand: $u = 4x + 1$

3. Differentiate to Find $dx$: Taking the derivative of $u$ with respect to $x$:

   \quad \Rightarrow \quad dx = \frac{du}{4}$$

4. Substituting in Terms of $u$: Substitute both $u$ and $dx$ into the original integral: $\int \sqrt{4x + 1} \, dx = \int \sqrt{u} \cdot \frac{du}{4}$ Simplify: $\frac{1}{4} \int u^{1/2} \, du$

5. Integrating: Use the power rule for integration: $\int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2u^{3/2}}{3}$ So, the integral becomes: $\frac{1}{4} \cdot \frac{2u^{3/2}}{3} = \frac{u^{3/2}}{6}$

6. Substituting Back $u = 4x + 1$: Finally, replace $u$ with $4x + 1$: $\frac{(4x + 1)^{3/2}}{6} + C$

This is the final solution to the integral.

Would you like any further explanation or clarification on a specific step?

Here are five related questions to expand on this topic:

1. How do you know when substitution is an appropriate technique for integration?
2. Could you use another method to solve this integral without substitution?
3. What is the power rule of integration, and how does it apply to integrals like this?
4. How would the solution change if the limits of integration were given?
5. What are some common functions where substitution is typically used?

Tip: Always check your substitution carefully to ensure that all parts of the original integral, including $dx$, are accounted for correctly.