`int ((1+sqrtx)^2)/sqrtx dx=`

Options

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

Answer

To solve the integral \(\int \frac{(1+\sqrt{x})^2}{\sqrt{x}} \, dx\), we will follow these steps:

### Step 1: Expand the integrand
First, we need to expand the expression \((1+\sqrt{x})^2\):
\[ (1+\sqrt{x})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{x} + (\sqrt{x})^2 = 1 + 2\sqrt{x} + x \]
Now, we can rewrite the integral:
\[ \int \frac{(1+\sqrt{x})^2}{\sqrt{x}} \, dx = \int \frac{1 + 2\sqrt{x} + x}{\sqrt{x}} \, dx \]

### Step 2: Simplify the integrand
Next, we divide each term by \(\sqrt{x}\):
\[ \int \left(\frac{1}{\sqrt{x}} + 2 + \sqrt{x}\right) \, dx \]

### Step 3: Rewrite the terms
Now we can rewrite \(\frac{1}{\sqrt{x}}\) and \(\sqrt{x}\) using exponents:
\[ \frac{1}{\sqrt{x}} = x^{-1/2} \quad \text{and} \quad \sqrt{x} = x^{1/2} \]
Thus, the integral becomes:
\[ \int \left(x^{-1/2} + 2 + x^{1/2}\right) \, dx \]

### Step 4: Integrate term by term
Now we can integrate each term separately:
1. \(\int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}\)
2. \(\int 2 \, dx = 2x\)
3. \(\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}\)

Putting it all together, we have:
\[ \int \left(x^{-1/2} + 2 + x^{1/2}\right) \, dx = 2x^{1/2} + 2x + \frac{2}{3} x^{3/2} + C \]

### Step 5: Write the final answer
Thus, the final answer is:
\[ \int \frac{(1+\sqrt{x})^2}{\sqrt{x}} \, dx = 2\sqrt{x} + 2x + \frac{2}{3} x^{3/2} + C \]

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