If `int_(0)^(1) x^(m) (1-x)^(n) dx= R int_(0)^(1) x^(n) (1-x)^(m) dx`, then

Options

A. R=1
B. `R= -1`
C. `R= 1//2`
D. None of these

Answer

To solve the equation

\[ \int_{0}^{1} x^{m} (1-x)^{n} \, dx = R \int_{0}^{1} x^{n} (1-x)^{m} \, dx, \]

we will follow these steps:

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

We know that

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

In our case, we can apply this property to the left-hand side integral:

\[ \int_{0}^{1} x^{m} (1-x)^{n} \, dx = \int_{0}^{1} (1-x)^{n} x^{m} \, dx. \]

### Step 2: Change the variable

Now, we can change the variable in the integral:

Let \( u = 1 - x \). Then, \( du = -dx \) and the limits change from \( x = 0 \) to \( x = 1 \) to \( u = 1 \) to \( u = 0 \). Thus,

\[ \int_{0}^{1} x^{m} (1-x)^{n} \, dx = \int_{1}^{0} (1-u)^{m} u^{n} (-du) = \int_{0}^{1} (1-u)^{m} u^{n} \, du. \]

### Step 3: Write the integral in terms of \( n \) and \( m \)

This means we can express the left-hand side as:

\[ \int_{0}^{1} (1-x)^{m} x^{n} \, dx. \]

### Step 4: Equate the two integrals

Now we have:

\[ \int_{0}^{1} x^{m} (1-x)^{n} \, dx = \int_{0}^{1} (1-x)^{m} x^{n} \, dx. \]

### Step 5: Set up the equation

Thus, we can rewrite our original equation as:

\[ \int_{0}^{1} x^{m} (1-x)^{n} \, dx = R \int_{0}^{1} x^{n} (1-x)^{m} \, dx. \]

### Step 6: Compare the two sides

Since both integrals are equal, we can conclude that:

\[ 1 = R. \]

### Conclusion

Therefore, the value of \( R \) is:

\[ R = 1. \]