https://file.aiquickdraw.com/mathbot/file/17293967966945l5s7gpr.jpg

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.

Find the indefinite integral \( \int \frac{x + 5}{\sqrt{x}} \, dx \) and check the result by differentiation.

Learn how to solve the indefinite integral \( \int \frac{x + 5}{\sqrt{x}} \, dx \) using the power rule for integration. This problem includes step-by-step calculations and a differentiation check to verify the solution.

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