The integral `I=int(e^(sqrtx)cos(e^(sqrtx)))/(sqrtx)dx=f(x)+c` (where, c is the constant of integration) and `f(ln((pi)/(4)))^(2)=sqrt2.` Then, the number of solutions of `f(x)=2e (AA x in R-{0})` is equal to

Answer

To solve the problem step by step, we need to evaluate the integral and find the function \( f(x) \) such that \( f( \ln( \frac{\pi}{4})^2 ) = \sqrt{2} \). We then need to determine the number of solutions for the equation \( f(x) = 2e \).

### Step 1: Evaluate the integral

We start with the integral:

\[ I = \int \frac{e^{\sqrt{x}} \cos(e^{\sqrt{x}})}{\sqrt{x}} \, dx \]

We will use the substitution \( t = e^{\sqrt{x}} \). To find \( dx \) in terms of \( dt \), we first express \( \sqrt{x} \) in terms of \( t \):

\[ \sqrt{x} = \ln(t) \quad \Rightarrow \quad x = (\ln(t))^2 \]

Next, we differentiate \( t = e^{\sqrt{x}} \):

\[ dt = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} \, dx \quad \Rightarrow \quad dx = 2\sqrt{x} \cdot \frac{dt}{e^{\sqrt{x}}} = 2\ln(t) \cdot \frac{dt}{t} \]

Now substituting \( t \) into the integral:

\[ I = \int \frac{t \cos(t)}{\ln(t)} \cdot 2\ln(t) \cdot \frac{dt}{t} = 2 \int \cos(t) \, dt \]

### Step 2: Integrate

The integral of \( \cos(t) \) is:

\[ \int \cos(t) \, dt = \sin(t) \]

Thus,

\[ I = 2\sin(t) + C \]

Substituting back \( t = e^{\sqrt{x}} \):

\[ I = 2\sin(e^{\sqrt{x}}) + C \]

### Step 3: Define the function \( f(x) \)

From the integral, we have:

\[ f(x) = 2\sin(e^{\sqrt{x}}) \]

### Step 4: Verify the condition

We need to verify that \( f(\ln(\frac{\pi}{4})^2) = \sqrt{2} \):

\[ f(\ln(\frac{\pi}{4})^2) = 2\sin(e^{\sqrt{\ln(\frac{\pi}{4})^2}}) = 2\sin(\frac{\pi}{4}) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \]

This condition is satisfied.

### Step 5: Solve for \( f(x) = 2e \)

Now we need to find the number of solutions for:

\[ 2\sin(e^{\sqrt{x}}) = 2e \]

Dividing both sides by 2 gives:

\[ \sin(e^{\sqrt{x}}) = e \]

### Step 6: Analyze the equation

The sine function \( \sin(y) \) has a range of \([-1, 1]\). Since \( e \approx 2.71 \), which is greater than 1, the equation \( \sin(e^{\sqrt{x}}) = e \) has no solutions.

### Conclusion

Thus, the number of solutions of \( f(x) = 2e \) is:

\[ \text{Number of solutions} = 0 \]