The value of `8/pi int_(0)^(pi/2)(cosx)^(2023)/((sinx)^(2023)+(cosx)^(2023))dx` is

Answer

To solve the integral

\[ I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} \, dx, \]

we can use a property of definite integrals.

### Step 1: Apply the substitution
Let’s denote the integral as \( I \). We can use the substitution \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \) and the limits change as follows: when \( x = 0 \), \( t = \frac{\pi}{2} \) and when \( x = \frac{\pi}{2} \), \( t = 0 \). Thus, we have:

\[ I = \frac{8}{\pi} \int_{\frac{\pi}{2}}^{0} \frac{(\cos(\frac{\pi}{2} - t))^{2023}}{(\sin(\frac{\pi}{2} - t))^{2023} + (\cos(\frac{\pi}{2} - t))^{2023}} (-dt). \]

### Step 2: Simplify the integral
Using the identities \( \cos(\frac{\pi}{2} - t) = \sin t \) and \( \sin(\frac{\pi}{2} - t) = \cos t \), we can rewrite the integral:

\[ I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\sin t)^{2023}}{(\cos t)^{2023} + (\sin t)^{2023}} \, dt. \]

### Step 3: Combine the two integrals
Now we have two expressions for \( I \):

\[ I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} \, dx, \]

and

\[ I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\sin x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} \, dx. \]

### Step 4: Add the two integrals
Adding these two expressions for \( I \):

\[ 2I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \left( \frac{(\cos x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} + \frac{(\sin x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} \right) dx. \]

The sum of the fractions simplifies to:

\[ 2I = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023} + (\sin x)^{2023}}{(\sin x)^{2023} + (\cos x)^{2023}} \, dx = \frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

### Step 5: Evaluate the integral
Now, we can evaluate the integral:

\[ 2I = \frac{8}{\pi} \cdot \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{8}{\pi} \cdot \frac{\pi}{2} = 4. \]

### Step 6: Solve for \( I \)
Thus, we have:

\[ 2I = 4 \implies I = 2. \]

### Final Answer
The value of the integral is

\[ \boxed{2}. \]