`int(root(3)x ^(2)+root(4)x+root(3)x)/(sqrtx)dx=`

Options

A. `(6x^((7)/(6)))/(7)+(4x^((4)/(3)))/(3)+(6x^((5)/(6)))/(5)+c`
B. `(7x^((7)/(6)))/(6)+(3x^((3)/(4)))/(4)+(5x^((5)/(6)))/(6)+c`
C. `(x^((7)/(6)))/(7)+(x^((3)/(4)))/(3)+(x^((5)/(6)))/(5)+c`
D. `(x^((7)/(6)))/(6)+(x^((3)/(4)))/(4)+(x^((5)/(6)))/(6)+c`

Answer

To solve the integral

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

we can simplify the expression step by step.

### Step 1: Simplify the integrand

First, we can combine the terms in the numerator:

\[ \sqrt{3}x^2 + \sqrt{4}x + \sqrt{3}x = \sqrt{3}x^2 + 2x + \sqrt{3}x = \sqrt{3}x^2 + (2 + \sqrt{3})x. \]

Now, we can rewrite the integral as:

\[ \int \frac{\sqrt{3}x^2 + (2 + \sqrt{3})x}{\sqrt{x}} \, dx. \]

### Step 2: Split the integral

We can split the integral into two separate integrals:

\[ \int \frac{\sqrt{3}x^2}{\sqrt{x}} \, dx + \int \frac{(2 + \sqrt{3})x}{\sqrt{x}} \, dx. \]

### Step 3: Simplify each term

Now, simplify each term:

1. For the first term:

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

2. For the second term:

\[ \frac{(2 + \sqrt{3})x}{\sqrt{x}} = (2 + \sqrt{3})x^{1 - \frac{1}{2}} = (2 + \sqrt{3})x^{\frac{1}{2}}. \]

### Step 4: Rewrite the integral

Now, we can rewrite the integral as:

\[ \int \sqrt{3}x^{\frac{3}{2}} \, dx + \int (2 + \sqrt{3})x^{\frac{1}{2}} \, dx. \]

### Step 5: Integrate each term

Now we can integrate each term separately:

1. For the first term:

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

2. For the second term:

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

### Step 6: Combine the results

Now, combine the results of the integrals:

\[ \int \frac{\sqrt{3}x^2 + (2 + \sqrt{3})x}{\sqrt{x}} \, dx = \frac{2\sqrt{3}}{5} x^{\frac{5}{2}} + \frac{2(2 + \sqrt{3})}{3} x^{\frac{3}{2}} + C, \]

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

### Final Answer

Thus, the final answer is:

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