If m,`n in N`, then
`int_(0)^(pi//2)((sin^(m)x)^(1/n))/((sin^(m)x)^(1/n)+(cos^(m)x)^(1/n))dx` is equal to

Options

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

Explanation

To solve the integral

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

we can use the property of definite integrals which states that

\[ \int_{0}^{b} f(x) \, dx = \int_{0}^{b} f(b - x) \, dx. \]

### Step 1: Apply the property of definite integrals

Let’s denote the integral as \( I \):

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

Now, we will apply the property:

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

### Step 2: Simplify using trigonometric identities

Using the identities:

\[ \sin\left(\frac{\pi}{2} - x\right) = \cos x \quad \text{and} \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x, \]

we can rewrite the integral:

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

### Step 3: Combine the two expressions for \( I \)

Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{(\sin^m x)^{\frac{1}{n}}}{(\sin^m x)^{\frac{1}{n}} + (\cos^m x)^{\frac{1}{n}}} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{(\cos^m x)^{\frac{1}{n}}}{(\cos^m x)^{\frac{1}{n}} + (\sin^m x)^{\frac{1}{n}}} \, dx \)

Adding these two equations gives:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{(\sin^m x)^{\frac{1}{n}} + (\cos^m x)^{\frac{1}{n}}}{(\sin^m x)^{\frac{1}{n}} + (\cos^m x)^{\frac{1}{n}}} \right) \, dx = \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

### Step 4: Evaluate the integral

The integral simplifies to:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

### Step 5: Solve for \( I \)

Dividing both sides by 2, we find:

\[ I = \frac{\pi}{4}. \]

### Final Answer

Thus, the value of the integral is:

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