The integral in the image is:

$\int \frac{x + 5}{\sqrt{x}} \, dx$

To solve this, let's first break it into two simpler integrals by rewriting the expression as follows:

$\int \frac{x}{\sqrt{x}} \, dx + \int \frac{5}{\sqrt{x}} \, dx$

We know that $\frac{x}{\sqrt{x}} = \sqrt{x} = x^{1/2}$, and $\frac{5}{\sqrt{x}} = 5x^{-1/2}$. So the integral becomes:

$\int x^{1/2} \, dx + 5 \int x^{-1/2} \, dx$

Now, solve each part separately:

1. For $\int x^{1/2} \, dx$, we use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ Applying this rule for $n = 1/2$: $\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$

2. For $5 \int x^{-1/2} \, dx$, apply the power rule again with $n = -1/2$: $\int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$ So: $5 \int x^{-1/2} \, dx = 5 \cdot 2x^{1/2} = 10x^{1/2}$

Putting everything together, we have the result:

$\int \frac{x + 5}{\sqrt{x}} \, dx = \frac{2}{3} x^{3/2} + 10x^{1/2} + C$

Differentiation Check:

To verify, we differentiate the result: $f(x) = \frac{2}{3} x^{3/2} + 10x^{1/2}$ Using the power rule for differentiation: $f'(x) = \frac{2}{3} \cdot \frac{3}{2} x^{1/2} + 10 \cdot \frac{1}{2} x^{-1/2} = x^{1/2} + 5x^{-1/2} = \frac{x + 5}{\sqrt{x}}$

Since the derivative matches the original integrand, the solution is correct.

Do you want more details or have any questions?

Here are 5 related questions:

1. How would you solve the integral if it involved a different fractional exponent?
2. What if there was a constant outside the entire integral?
3. Can the power rule be applied to all integrals with $x^n$?
4. What would happen if the integrand included terms like $x^{3/2}$?
5. How does the constant of integration affect the result of indefinite integrals?

Tip: Always verify your integrals by differentiating the result to ensure correctness.