`int (x^(2) +2x -5)/(sqrt(x))dx`

Answer

To solve the integral \(\int \frac{x^2 + 2x - 5}{\sqrt{x}} \, dx\), we will break it down step by step.

### Step 1: Rewrite the Integral
We start by separating the terms in the numerator and rewriting the integral:
\[ \int \frac{x^2}{\sqrt{x}} \, dx + \int \frac{2x}{\sqrt{x}} \, dx - \int \frac{5}{\sqrt{x}} \, dx \]

### Step 2: Simplify Each Term
Next, we simplify each term:
1. \(\frac{x^2}{\sqrt{x}} = x^{2 - \frac{1}{2}} = x^{\frac{3}{2}}\)
2. \(\frac{2x}{\sqrt{x}} = 2 \cdot x^{1 - \frac{1}{2}} = 2 \cdot x^{\frac{1}{2}}\)
3. \(\frac{5}{\sqrt{x}} = 5 \cdot x^{-\frac{1}{2}}\)

Thus, we can rewrite the integral as:
\[ \int x^{\frac{3}{2}} \, dx + 2 \int x^{\frac{1}{2}} \, dx - 5 \int x^{-\frac{1}{2}} \, dx \]

### Step 3: Integrate Each Term
Now we will integrate each term using the formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\):
1. \(\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}\)
2. \(2 \int x^{\frac{1}{2}} \, dx = 2 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = 2 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{4}{3} x^{\frac{3}{2}}\)
3. \(-5 \int x^{-\frac{1}{2}} \, dx = -5 \cdot \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = -5 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -10 x^{\frac{1}{2}}\)

### Step 4: Combine the Results
Now we combine all the results:
\[ \frac{2}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} - 10 x^{\frac{1}{2}} + C \]

### Final Answer
Thus, the final answer for the integral is:
\[ \int \frac{x^2 + 2x - 5}{\sqrt{x}} \, dx = \frac{2}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} - 10 x^{\frac{1}{2}} + C \]