https://file.aiquickdraw.com/mathbot/file/172650541322755cacsts.jpg

The integral presented in the image is:

\[
\int \frac{\sqrt{x}}{\sqrt{x} - 3} \, dx
\]

### Approach:
To solve this integral, one common technique is to use substitution. Here's the step-by-step solution:

#### Step 1: Substitution
Let:
\[
u = \sqrt{x}
\]
This implies:
\[
x = u^2 \quad \text{and} \quad dx = 2u \, du
\]

#### Step 2: Rewrite the Integral
Substitute \( \sqrt{x} = u \) and \( dx = 2u \, du \) into the integral:
\[
\int \frac{\sqrt{x}}{\sqrt{x} - 3} \, dx = \int \frac{u}{u - 3} \cdot 2u \, du
\]
This simplifies to:
\[
2 \int \frac{u^2}{u - 3} \, du
\]

#### Step 3: Perform Long Division
Now, perform long division on \( \frac{u^2}{u - 3} \):
\[
\frac{u^2}{u - 3} = u + 3 + \frac{9}{u - 3}
\]

#### Step 4: Rewrite the Integral
Now, the integral becomes:
\[
2 \int \left( u + 3 + \frac{9}{u - 3} \right) du
\]
Breaking it up:
\[
2 \left( \int u \, du + 3 \int du + 9 \int \frac{1}{u - 3} \, du \right)
\]

#### Step 5: Evaluate Each Term
1. \( \int u \, du = \frac{u^2}{2} \)
2. \( \int du = u \)
3. \( \int \frac{1}{u - 3} \, du = \ln |u - 3| \)

So, the full integral becomes:
\[
2 \left( \frac{u^2}{2} + 3u + 9 \ln |u - 3| \right) = u^2 + 6u + 18 \ln |u - 3|
\]

#### Step 6: Substitute Back \( u = \sqrt{x} \)
Finally, substitute \( u = \sqrt{x} \) back into the result:
\[
\sqrt{x}^2 + 6\sqrt{x} + 18 \ln |\sqrt{x} - 3|
\]
Simplifying:
\[
x + 6\sqrt{x} + 18 \ln |\sqrt{x} - 3| + C
\]

Thus, the solution to the integral is:

\[
\boxed{x + 6\sqrt{x} + 18 \ln |\sqrt{x} - 3| + C}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. What other substitution methods could work for integrals like this?
2. How can partial fraction decomposition be applied in similar problems?
3. Can you solve this integral using a different technique like integration by parts?
4. What are the conditions for using logarithmic functions in the solution?
5. How would the presence of a higher-order root, like \( \sqrt[3]{x} \), affect the approach?

**Tip**: Long division can simplify integrals involving polynomials in the numerator and denominator, breaking them into easier terms.

Evaluate the integral: ∫ (sqrt(x) / (sqrt(x) - 3)) dx.

Solve the integral ∫ (sqrt(x) / (sqrt(x) - 3)) dx using substitution and long division. This solution explains step-by-step integration techniques, including the application of logarithmic functions and simplifications through algebraic methods.

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