If the antiderivative of `(1)/(sqrt(x+x^(3//2)))` is `Asqrt(1+sqrt(x))+C` then A is equal to

Options

A. 2
B. 1
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( A \) in the expression for the antiderivative of the function \( \frac{1}{\sqrt{x + x^{3/2}}} \).

### Step-by-Step Solution:

1. **Set up the integral**:
We need to evaluate the integral:
\[ I = \int \frac{1}{\sqrt{x + x^{3/2}}} \, dx \]

2. **Simplify the integrand**:
We can factor out \( \sqrt{x} \) from the expression under the square root:
\[ x + x^{3/2} = x(1 + \sqrt{x}) \]
Therefore, we have:
\[ I = \int \frac{1}{\sqrt{x(1 + \sqrt{x})}} \, dx \]

3. **Substitution**:
Let \( t = 1 + \sqrt{x} \). Then, we differentiate to find \( dx \):
\[ \sqrt{x} = t - 1 \implies x = (t - 1)^2 \]
Differentiating gives:
\[ dx = 2(t - 1) \, dt \]

4. **Change of variables**:
Substitute \( x \) and \( dx \) into the integral:
\[ I = \int \frac{2(t - 1)}{\sqrt{(t - 1)^2} \sqrt{t}} \, dt \]
Since \( \sqrt{(t - 1)^2} = t - 1 \) (for \( t > 1 \)), we get:
\[ I = \int \frac{2(t - 1)}{(t - 1) \sqrt{t}} \, dt = \int \frac{2}{\sqrt{t}} \, dt \]

5. **Integrate**:
The integral simplifies to:
\[ I = 2 \int t^{-1/2} \, dt = 2 \cdot 2t^{1/2} + C = 4\sqrt{t} + C \]

6. **Back substitute**:
Now substitute back \( t = 1 + \sqrt{x} \):
\[ I = 4\sqrt{1 + \sqrt{x}} + C \]

7. **Compare with the given form**:
We are given that the antiderivative can be expressed as:
\[ A\sqrt{1 + \sqrt{x}} + C \]
By comparing, we see that:
\[ A = 4 \]

### Final Answer:
Thus, the value of \( A \) is:
\[ \boxed{4} \]