The integral provided in the image is:

$\int \frac{\sqrt{x} + 4}{x} \, dx$

Step-by-Step Solution:

1. Split the Integral: Rewrite the expression as two separate terms: $\int \frac{\sqrt{x} + 4}{x} \, dx = \int \frac{\sqrt{x}}{x} \, dx + \int \frac{4}{x} \, dx$

2. Simplify Each Term:

   • For the first term: $\frac{\sqrt{x}}{x} = \frac{x^{1/2}}{x^1} = x^{-1/2}$ So the integral becomes: $\int x^{-1/2} \, dx$ Use the power rule of integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1$ Here $n = -1/2$: $\int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2\sqrt{x}$

   • For the second term: $\int \frac{4}{x} \, dx = 4 \int \frac{1}{x} \, dx = 4 \ln|x|$

3. Combine Results: Combine the results of the two integrals: $\int \frac{\sqrt{x} + 4}{x} \, dx = 2\sqrt{x} + 4 \ln|x| + C$

Final Answer:

$\boxed{2\sqrt{x} + 4 \ln|x| + C}$

Let me know if you want further clarification or if you have additional questions!

Related Questions:

1. What is the power rule of integration, and how is it applied in this problem?
2. How do you handle integrals involving logarithms, like $\int \frac{1}{x} dx$?
3. How would the solution change if the integrand were $\frac{\sqrt{x} - 4}{x}$?
4. Can you generalize the integration process for $\int \frac{\sqrt{x} + k}{x} dx$, where $k$ is a constant?
5. What are the domain restrictions for the function $\ln|x|$ in the integral?

Tip:

Always split integrals into simpler terms when dealing with sums or quotients, as it makes applying basic rules of integration more straightforward.