Find the integral `I= int( x^(2) + 5x-1) /( sqrt(x) ) dx`.

Answer

To solve the integral \( I = \int \frac{x^2 + 5x - 1}{\sqrt{x}} \, dx \), we can start by simplifying the integrand.

### Step 1: Rewrite the integrand
We can separate the integrand into simpler terms:
\[ I = \int \left( \frac{x^2}{\sqrt{x}} + \frac{5x}{\sqrt{x}} - \frac{1}{\sqrt{x}} \right) \, dx \]
This simplifies to:
\[ I = \int \left( x^{2 - \frac{1}{2}} + 5x^{1 - \frac{1}{2}} - x^{-\frac{1}{2}} \right) \, dx \]
Which can be rewritten as:
\[ I = \int \left( x^{\frac{3}{2}} + 5x^{\frac{1}{2}} - x^{-\frac{1}{2}} \right) \, dx \]

### Step 2: Integrate each term
Now we can integrate each term separately:
1. For \( \int x^{\frac{3}{2}} \, dx \):
\[ \int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}} \]

2. For \( \int 5x^{\frac{1}{2}} \, dx \):
\[ \int 5x^{\frac{1}{2}} \, dx = 5 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = 5 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{10}{3} x^{\frac{3}{2}} \]

3. For \( \int -x^{-\frac{1}{2}} \, dx \):
\[ \int -x^{-\frac{1}{2}} \, dx = -\frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = -\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -2x^{\frac{1}{2}} \]

### Step 3: Combine the results
Now we can combine all the results:
\[ I = \frac{2}{5} x^{\frac{5}{2}} + \frac{10}{3} x^{\frac{3}{2}} - 2x^{\frac{1}{2}} + C \]
where \( C \) is the constant of integration.

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