The value of the definite integral
`int_(t+2pi)^(t+5pi//2) {sin^(-1)(cosx)+cos^(-1)(cosx)}`dx is equal to

Options

A. `(pi^(2))/(2)`
B. `(pi^(2))/(8)`
C. `(pi^(2))/(4)`
D. none of these

Explanation

To solve the definite integral

\[ I = \int_{t + 2\pi}^{t + \frac{5\pi}{2}} \left( \sin^{-1}(\cos x) + \cos^{-1}(\cos x) \right) \, dx, \]

we can use the identity:

\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}. \]

### Step 1: Apply the Identity
Using the identity, we can rewrite the integrand:

\[ \sin^{-1}(\cos x) + \cos^{-1}(\cos x) = \frac{\pi}{2}. \]

Thus, the integral simplifies to:

\[ I = \int_{t + 2\pi}^{t + \frac{5\pi}{2}} \frac{\pi}{2} \, dx. \]

### Step 2: Evaluate the Integral
Since \(\frac{\pi}{2}\) is a constant, we can factor it out of the integral:

\[ I = \frac{\pi}{2} \int_{t + 2\pi}^{t + \frac{5\pi}{2}} 1 \, dx. \]

The integral of 1 over the interval \([t + 2\pi, t + \frac{5\pi}{2}]\) is simply the length of the interval:

\[ \int_{t + 2\pi}^{t + \frac{5\pi}{2}} 1 \, dx = \left( t + \frac{5\pi}{2} \right) - \left( t + 2\pi \right) = \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}. \]

### Step 3: Substitute Back
Now substituting this back into the expression for \(I\):

\[ I = \frac{\pi}{2} \cdot \frac{\pi}{2} = \frac{\pi^2}{4}. \]

### Final Result
Thus, the value of the definite integral is

\[ \boxed{\frac{\pi^2}{4}}. \]

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