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

Options

A. `(pi^(2))/(8)`
B. `(pi^(2))/(4)`
C. `(pi^(2))/(2)`
D. `pi^(2)`

Explanation

To solve the definite integral

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

we can start by simplifying the expression inside the integral.

### Step 1: Simplify the expression inside the integral

We know that:

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

This identity holds true because for any angle \(x\):

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

Thus, we can rewrite our integral as:

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

### Step 2: Evaluate the integral

Now we can evaluate the integral:

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

The integral of \(dx\) from \(2\pi\) to \(\frac{5\pi}{2}\) is simply the length of the interval:

\[ \int_{2\pi}^{\frac{5\pi}{2}} dx = \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}. \]

### Step 3: Substitute back to find \(I\)

Now substitute this result back into our expression for \(I\):

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

Thus, the value of the definite integral is:

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