https://file.aiquickdraw.com/mathbot/file/1725704515275zua8xbyk.jpg

The integral in the image is:

\[
\int \frac{x+2}{x \sqrt{x - 3}} \, dx
\]

### Step-by-step solution:

1. **Substitute**: A good substitution would be \( u = x - 3 \), which simplifies the square root and the expression inside. Let's proceed with this:

   \[
   u = x - 3 \quad \Rightarrow \quad du = dx
   \]

   Also, \( x = u + 3 \).

2. **Rewrite the integral**: Substitute these into the original integral:

   \[
   \int \frac{(u + 3) + 2}{(u + 3) \sqrt{u}} \, du
   \]

   Simplifying the numerator:

   \[
   = \int \frac{u + 5}{(u + 3) \sqrt{u}} \, du
   \]

3. **Separate terms**: Split the integral into two parts:

   \[
   \int \frac{u}{(u + 3) \sqrt{u}} \, du + \int \frac{5}{(u + 3) \sqrt{u}} \, du
   \]

   Simplify each term further and solve accordingly, using standard integration techniques, like partial fractions and substitution for the second term.

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**Tip:** Substitutions in integrals are a powerful tool when dealing with square roots or complex expressions. Choosing the right substitution simplifies the problem greatly.

Solve the integral ∫(x + 2) / [x√(x - 3)] dx using substitution (u = x - 3) and partial fractions. This step-by-step solution is designed for students at the undergraduate calculus level.

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