`int x sqrt(x+2)dx` is equal to

Options

A. `(2)/(5)(x+2)^(5//2)-(4)/(3)(x+2)^(3//2)+C`
B. `(1)/(5)(x+2)^(5//2)-(4)/(3)(x+2)^(3//2)+C`
C. `(2)/(5)(x+2)^(5//2)-(2)/(3)(x+2)^(3//2)+C`
D. None of these

Explanation

To solve the integral \( I = \int x \sqrt{x+2} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int x \sqrt{x+2} \, dx \]
We can express \( x \) in terms of \( (x + 2) \):
\[ x = (x + 2) - 2 \]
Thus, we can rewrite the integral as:
\[ I = \int ((x + 2) - 2) \sqrt{x+2} \, dx \]
This simplifies to:
\[ I = \int (x + 2) \sqrt{x+2} \, dx - 2 \int \sqrt{x+2} \, dx \]

### Step 2: Change of Variable
Let \( u = x + 2 \). Then, \( du = dx \) and \( x = u - 2 \). The integral becomes:
\[ I = \int u \sqrt{u} \, du - 2 \int \sqrt{u} \, du \]
This can be simplified to:
\[ I = \int u^{3/2} \, du - 2 \int u^{1/2} \, du \]

### Step 3: Integrate Each Term
Now we can integrate each term separately:
1. For \( \int u^{3/2} \, du \):
\[ \int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2} \]
2. For \( \int u^{1/2} \, du \):
\[ \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \]

### Step 4: Substitute Back
Substituting back into our expression for \( I \):
\[ I = \frac{2}{5} u^{5/2} - 2 \cdot \frac{2}{3} u^{3/2} \]
This simplifies to:
\[ I = \frac{2}{5} u^{5/2} - \frac{4}{3} u^{3/2} \]

### Step 5: Substitute \( u \) Back to \( x \)
Now substitute \( u = x + 2 \) back into the equation:
\[ I = \frac{2}{5} (x + 2)^{5/2} - \frac{4}{3} (x + 2)^{3/2} + C \]

### Final Answer
Thus, the integral \( \int x \sqrt{x+2} \, dx \) is:
\[ I = \frac{2}{5} (x + 2)^{5/2} - \frac{4}{3} (x + 2)^{3/2} + C \]