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

Options

A. `x+12/7x^(7/6)+3/4x^(4/3)+c`
B. `(2x^(3/2))/3+(3x^(7/3))/7+c`
C. `x+4/5x^(5/6)+c`
D. `x+12/5x^(5/6)+c`

Answer

To solve the integral \( \int \frac{(\sqrt{x} + \sqrt[3]{x^2})^2}{x} \, dx \), we will follow these steps:

### Step 1: Expand the expression
First, we expand the expression \( (\sqrt{x} + \sqrt[3]{x^2})^2 \) using the formula \( (a + b)^2 = a^2 + b^2 + 2ab \).

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

Calculating each term:
- \( (\sqrt{x})^2 = x \)
- \( (\sqrt[3]{x^2})^2 = x^{\frac{2}{3}} \)
- \( 2(\sqrt{x})(\sqrt[3]{x^2}) = 2 \cdot x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} = 2x^{\frac{1}{2} + \frac{2}{3}} = 2x^{\frac{3}{6} + \frac{4}{6}} = 2x^{\frac{7}{6}} \)

Thus, we have:
\[ (\sqrt{x} + \sqrt[3]{x^2})^2 = x + x^{\frac{2}{3}} + 2x^{\frac{7}{6}} \]

### Step 2: Rewrite the integral
Now, substitute this back into the integral:
\[ \int \frac{x + x^{\frac{2}{3}} + 2x^{\frac{7}{6}}}{x} \, dx \]

This simplifies to:
\[ \int \left(1 + x^{-\frac{1}{3}} + 2x^{\frac{1}{6}}\right) \, dx \]

### Step 3: Integrate term by term
Now we can integrate each term separately:
1. The integral of \( 1 \) is \( x \).
2. The integral of \( x^{-\frac{1}{3}} \) is:
\[ \frac{x^{-\frac{1}{3} + 1}}{-\frac{1}{3} + 1} = \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = \frac{3}{2} x^{\frac{2}{3}} \]
3. The integral of \( 2x^{\frac{1}{6}} \) is:
\[ 2 \cdot \frac{x^{\frac{1}{6} + 1}}{\frac{1}{6} + 1} = 2 \cdot \frac{x^{\frac{7}{6}}}{\frac{7}{6}} = \frac{12}{7} x^{\frac{7}{6}} \]

### Step 4: Combine the results
Combining all the results, we have:
\[ \int \frac{(\sqrt{x} + \sqrt[3]{x^2})^2}{x} \, dx = x + \frac{3}{2} x^{\frac{2}{3}} + \frac{12}{7} x^{\frac{7}{6}} + C \]

### Final Answer
Thus, the final answer is:
\[ x + \frac{3}{2} x^{\frac{2}{3}} + \frac{12}{7} x^{\frac{7}{6}} + C \]