`int (2)/(sqrtx-sqrt(x+3)) dx`=….

Options

A. `-(2)/(3) [x^((3)/(2)) + (x+3)^((3)/(2))]+c`
B. `(2)/(3) [x^((3)/(2))-(x+3)^((3)/(2))]+c`
C. `(4)/(9) [x^((3)/(2)) -(x+3)^((3)/(2))]+c`
D. `-(4)/(9) [x^((3)/(2)) + (x+ 3)^((3)/(2))]+c`

Answer

To solve the integral \( \int \frac{2}{\sqrt{x} - \sqrt{x+3}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
Let \( I = \int \frac{2}{\sqrt{x} - \sqrt{x+3}} \, dx \).

### Step 2: Rationalize the Denominator
To simplify the integral, we can multiply the numerator and the denominator by the conjugate of the denominator:

\[ I = \int \frac{2(\sqrt{x} + \sqrt{x+3})}{(\sqrt{x} - \sqrt{x+3})(\sqrt{x} + \sqrt{x+3})} \, dx \]

### Step 3: Simplify the Denominator
The denominator simplifies as follows:

\[ (\sqrt{x} - \sqrt{x+3})(\sqrt{x} + \sqrt{x+3}) = x - (x + 3) = -3 \]

So, we have:

\[ I = \int \frac{2(\sqrt{x} + \sqrt{x+3})}{-3} \, dx = -\frac{2}{3} \int (\sqrt{x} + \sqrt{x+3}) \, dx \]

### Step 4: Split the Integral
Now, we can split the integral:

\[ I = -\frac{2}{3} \left( \int \sqrt{x} \, dx + \int \sqrt{x+3} \, dx \right) \]

### Step 5: Integrate Each Term
Now we will integrate each term separately.

1. For \( \int \sqrt{x} \, dx \):

\[ \int \sqrt{x} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]

2. For \( \int \sqrt{x+3} \, dx \):

Using the substitution \( u = x + 3 \), \( du = dx \):

\[ \int \sqrt{x+3} \, dx = \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} = \frac{2}{3} (x + 3)^{3/2} \]

### Step 6: Combine the Results
Now, substituting back into the integral:

\[ I = -\frac{2}{3} \left( \frac{2}{3} x^{3/2} + \frac{2}{3} (x + 3)^{3/2} \right) \]

### Step 7: Simplify
This simplifies to:

\[ I = -\frac{4}{9} \left( x^{3/2} + (x + 3)^{3/2} \right) + C \]

### Final Answer
Thus, the final result is:

\[ \int \frac{2}{\sqrt{x} - \sqrt{x+3}} \, dx = -\frac{4}{9} \left( x^{3/2} + (x + 3)^{3/2} \right) + C \]

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