`"Let " f(x)=int x^(sinx)(1+xcosx*In x+sinx)dx " and " f((pi)/(2))=(pi^(2))/(4). " Then the value of " cos f(pi) " is"- .`

Explanation

To solve the problem step by step, we will analyze the function \( f(x) \) given by the integral:

\[ f(x) = \int x^{\sin x} (1 + x \cos x \ln x + \sin x) \, dx \]

### Step 1: Identify the integral
We start with the integral expression for \( f(x) \):

\[ f(x) = \int x^{\sin x} (1 + x \cos x \ln x + \sin x) \, dx \]

### Step 2: Simplify the integral
Notice that the integrand can be expressed in a more manageable form. We can rewrite the first part of the integrand:

\[ f(x) = \int x^{\sin x} \, dx + \int x^{\sin x} (x \cos x \ln x + \sin x) \, dx \]

### Step 3: Use substitution
Let \( F(x) = x^{\sin x} \). Then, we can differentiate \( F(x) \):

\[ F'(x) = e^{\sin x \ln x} \left( \sin x \cdot \frac{1}{x} + \ln x \cdot \cos x \right) \]

### Step 4: Integrate by parts
The integral can be expressed as:

\[ f(x) = \int F'(x) \, dx \]

Using integration by parts, we can express \( f(x) \) in terms of \( F(x) \):

\[ f(x) = x F(x) - \int x F'(x) \, dx \]

### Step 5: Evaluate \( f\left(\frac{\pi}{2}\right) \)
We know from the problem statement that:

\[ f\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4} \]

Substituting \( x = \frac{\pi}{2} \):

\[ f\left(\frac{\pi}{2}\right) = \frac{\pi}{2} \left(\frac{\pi}{2}\right)^{\sin\left(\frac{\pi}{2}\right)} + C = \frac{\pi}{2} \cdot \frac{\pi}{2} + C = \frac{\pi^2}{4} + C \]

This gives us \( C = 0 \).

### Step 6: Calculate \( f(\pi) \)
Now we need to find \( f(\pi) \):

\[ f(\pi) = \pi \cdot \pi^{\sin(\pi)} + C \]

Since \( \sin(\pi) = 0 \):

\[ f(\pi) = \pi \cdot \pi^0 + 0 = \pi \]

### Step 7: Find \( \cos(f(\pi)) \)
Now we calculate \( \cos(f(\pi)) \):

\[ \cos(f(\pi)) = \cos(\pi) = -1 \]

### Final Answer
Thus, the value of \( \cos(f(\pi)) \) is:

\[ \boxed{-1} \]