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 given integral problem step by step, we start with the integral:

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

We know that this integral can be expressed in the form \( I = f(x) + c \), where \( c \) is the constant of integration. We also have the condition that \( f(\ln(\frac{\pi}{4}))^2 = \sqrt{2} \).

### Step 1: Substitution
Let’s make the substitution:
\[ t = e^{\sqrt{x}} \]
Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{2\sqrt{x}} e^{\sqrt{x}} \implies dx = 2\sqrt{x} \frac{dt}{e^{\sqrt{x}}} \]
Since \( t = e^{\sqrt{x}} \), we can express \( \sqrt{x} \) in terms of \( t \):
\[ \sqrt{x} = \ln(t) \]
Thus, we can rewrite \( dx \) as:
\[ dx = 2 \ln(t) \frac{dt}{t} \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ I = \int \frac{t \cos(t)}{\ln(t)} \cdot 2 \ln(t) \frac{dt}{t} = 2 \int \cos(t) \, dt \]
This simplifies to:
\[ I = 2 \sin(t) + c \]

### Step 3: Back Substitute
Now, substituting back for \( t \):
\[ I = 2 \sin(e^{\sqrt{x}}) + c \]
Thus, we have:
\[ f(x) = 2 \sin(e^{\sqrt{x}}) \]

### Step 4: Use the Given Condition
We know that:
\[ f(\ln(\frac{\pi}{4}))^2 = \sqrt{2} \]
Calculating \( f(\ln(\frac{\pi}{4})) \):
\[ f(\ln(\frac{\pi}{4})) = 2 \sin(e^{\sqrt{\ln(\frac{\pi}{4})}}) \]
Now, simplifying \( \sqrt{\ln(\frac{\pi}{4})} \):
\[ \sqrt{\ln(\frac{\pi}{4})} = \sqrt{\ln(\pi) - \ln(4)} = \sqrt{\ln(\pi) - 2\ln(2)} = \sqrt{\ln(\frac{\pi}{4})} \]
Thus,
\[ f(\ln(\frac{\pi}{4})) = 2 \sin(\frac{\pi}{4}) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \]
This confirms the condition.

### Step 5: Solve for \( f(x) = 2e \)
Now we need to find the number of solutions to the equation:
\[ f(x) = 2e \]
This translates to:
\[ 2 \sin(e^{\sqrt{x}}) = 2e \implies \sin(e^{\sqrt{x}}) = e \]
Since the sine function has a range of \([-1, 1]\), and \( e \approx 2.718\) is outside this range, there are no solutions to this equation.

### Conclusion
Thus, the number of solutions of \( f(x) = 2e \) is:
\[ \text{Number of solutions} = 0 \]