Evaluate the following integrals.
`int (x^(2) + 3x^(2) + 4)/(sqrt(x))dx`

Answer

To evaluate the integral

\[ \int \frac{x^2 + 3x^2 + 4}{\sqrt{x}} \, dx, \]

we can simplify the expression inside the integral first.

### Step 1: Simplify the integrand

Combine the terms in the numerator:

\[ x^2 + 3x^2 = 4x^2. \]

Thus, we can rewrite the integral as:

\[ \int \frac{4x^2 + 4}{\sqrt{x}} \, dx. \]

### Step 2: Split the integral

Now, we can split the integral into two parts:

\[ \int \frac{4x^2}{\sqrt{x}} \, dx + \int \frac{4}{\sqrt{x}} \, dx. \]

### Step 3: Simplify each term

For the first term, we simplify \(\frac{4x^2}{\sqrt{x}}\):

\[ \frac{4x^2}{\sqrt{x}} = 4x^{2 - \frac{1}{2}} = 4x^{\frac{3}{2}}. \]

For the second term, we simplify \(\frac{4}{\sqrt{x}}\):

\[ \frac{4}{\sqrt{x}} = 4x^{-\frac{1}{2}}. \]

### Step 4: Rewrite the integral

Now, we can rewrite the integral as:

\[ \int 4x^{\frac{3}{2}} \, dx + \int 4x^{-\frac{1}{2}} \, dx. \]

### Step 5: Integrate each term

Now we can integrate each term separately.

1. For the first term:

\[ \int 4x^{\frac{3}{2}} \, dx = 4 \cdot \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = 4 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{8}{5} x^{\frac{5}{2}}. \]

2. For the second term:

\[ \int 4x^{-\frac{1}{2}} \, dx = 4 \cdot \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = 4 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 8x^{\frac{1}{2}}. \]

### Step 6: Combine the results

Combining both results, we have:

\[ \frac{8}{5} x^{\frac{5}{2}} + 8x^{\frac{1}{2}} + C, \]

where \(C\) is the constant of integration.

### Final Answer

Thus, the final result of the integral is:

\[ \int \frac{x^2 + 3x^2 + 4}{\sqrt{x}} \, dx = \frac{8}{5} x^{\frac{5}{2}} + 8x^{\frac{1}{2}} + C. \]

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