If `f(x)=x+sinx`, then find `(2)/(pi^(2)).int_(pi)^(2pi)(f^(-1)(x)+sinx)dx`.

Explanation

To solve the problem, we need to evaluate the integral

\[ \frac{2}{\pi^2} \int_{\pi}^{2\pi} \left( f^{-1}(x) + \sin x \right) dx \]

where \( f(x) = x + \sin x \).

### Step 1: Understand the function and its inverse

First, we need to express \( f^{-1}(x) \). Since \( f(x) = x + \sin x \), we can set \( y = f(t) = t + \sin t \). This means \( t = f^{-1}(y) \).

### Step 2: Differentiate \( f(t) \)

To find \( dx \) in terms of \( dt \), we differentiate \( f(t) \):

\[ f'(t) = 1 + \cos t \]

Thus,

\[ dx = f'(t) dt = (1 + \cos t) dt \]

### Step 3: Change the limits of integration

Next, we need to change the limits of integration from \( x \) to \( t \). When \( x = \pi \):

\[ t = f^{-1}(\pi) \implies f(t) = \pi \implies t + \sin t = \pi \]

When \( x = 2\pi \):

\[ t = f^{-1}(2\pi) \implies f(t) = 2\pi \implies t + \sin t = 2\pi \]

### Step 4: Set up the integral

Now we can rewrite the integral:

\[ \int_{\pi}^{2\pi} (f^{-1}(x) + \sin x) dx = \int_{t_1}^{t_2} (t + \sin(f(t))) f'(t) dt \]

where \( t_1 = f^{-1}(\pi) \) and \( t_2 = f^{-1}(2\pi) \).

### Step 5: Evaluate the integral

We can split the integral:

\[ \int_{t_1}^{t_2} t f'(t) dt + \int_{t_1}^{t_2} \sin(f(t)) f'(t) dt \]

Using integration by parts on the first integral:

\[ \int t f'(t) dt = t f(t) - \int f(t) dt \]

Now we apply the limits \( t_1 \) to \( t_2 \):

\[ \left[ t f(t) \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} f(t) dt \]

### Step 6: Substitute back into the original integral

Now we substitute back into the original integral:

\[ \frac{2}{\pi^2} \left( \left[ t f(t) \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} f(t) dt + \int_{\pi}^{2\pi} \sin x dx \right) \]

### Step 7: Calculate the definite integrals

Calculate \( \int_{\pi}^{2\pi} \sin x dx \):

\[ \int \sin x dx = -\cos x \quad \Rightarrow \quad \left[-\cos x\right]_{\pi}^{2\pi} = -\cos(2\pi) + \cos(\pi) = -1 + 1 = 0 \]

### Step 8: Final evaluation

After substituting and simplifying, we find that the final result is:

\[ \frac{2}{\pi^2} \cdot 3 \frac{\pi^2}{2} = 3 \]

Thus, the required value is

\[ \boxed{3} \]