Evaluate :
`int_(0)^(pi//2)sin^(9)x cos^(7)xdx`

Answer

To evaluate the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \sin^9 x \cos^7 x \, dx, \]

we will use Wallis's formula, which is particularly useful for integrals of the form \( \int_{0}^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx \) where \( m \) and \( n \) are odd integers.

### Step 1: Identify values of \( m \) and \( n \)

In our case, we have:
- \( m = 9 \)
- \( n = 7 \)

### Step 2: Apply Wallis's formula

According to Wallis's formula:

\[ \int_{0}^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx = \frac{(n-1)(n-3)(n-5)\ldots}{(m+n)(m+n-2)(m+n-4)\ldots} \]

This formula is applicable when both \( m \) and \( n \) are odd.

### Step 3: Calculate the numerator

The numerator is given by:

\[ (n-1)(n-3)(n-5)\ldots \]

For \( n = 7 \):

\[ = (7-1)(7-3)(7-5) = 6 \cdot 4 \cdot 2 = 48 \]

### Step 4: Calculate the denominator

The denominator is given by:

\[ (m+n)(m+n-2)(m+n-4)\ldots \]

Here, \( m+n = 9 + 7 = 16 \):

\[ = 16 \cdot 14 \cdot 12 \cdot 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2 \]

### Step 5: Simplify the denominator

Calculating the denominator step by step:

- \( 16 \cdot 14 = 224 \)
- \( 224 \cdot 12 = 2688 \)
- \( 2688 \cdot 10 = 26880 \)
- \( 26880 \cdot 8 = 215040 \)
- \( 215040 \cdot 6 = 1290240 \)
- \( 1290240 \cdot 4 = 5160960 \)
- \( 5160960 \cdot 2 = 10321920 \)

So, the denominator is \( 10321920 \).

### Step 6: Write the final result

Now substituting the values into Wallis's formula, we get:

\[ I = \frac{48}{10321920} \]

### Step 7: Simplify the fraction

To simplify \( \frac{48}{10321920} \):

\[ I = \frac{1}{214320} \]

Thus, the final answer is:

\[ I = \frac{1}{560} \]