The value of `int_(0)^(1) x^(2) (1- x)^(9) dx` is

Options

A. `1/610`
B. `1/630`
C. `1/640`
D. `1/660`

Answer

To solve the integral \( I = \int_{0}^{1} x^2 (1 - x)^9 \, dx \), we can use the properties of definite integrals and the binomial theorem. Here’s a step-by-step solution:

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int_{0}^{1} x^2 (1 - x)^9 \, dx \]

### Step 2: Use the Beta Function
The integral can be expressed in terms of the Beta function:
\[ I = \int_{0}^{1} x^{a-1} (1 - x)^{b-1} \, dx = B(a, b) \]
where \( a = 3 \) and \( b = 10 \). The Beta function is related to the Gamma function:
\[ B(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a + b)} \]

### Step 3: Calculate the Gamma Functions
We need to calculate:
\[ \Gamma(3) = (3-1)! = 2! = 2 \]
\[ \Gamma(10) = (10-1)! = 9! = 362880 \]
\[ \Gamma(13) = (13-1)! = 12! = 479001600 \]

### Step 4: Substitute into the Beta Function
Now we substitute into the Beta function:
\[ I = B(3, 10) = \frac{\Gamma(3) \Gamma(10)}{\Gamma(13)} = \frac{2 \times 362880}{479001600} \]

### Step 5: Simplify the Expression
Calculating the numerator:
\[ 2 \times 362880 = 725760 \]
Now substitute this back:
\[ I = \frac{725760}{479001600} \]

### Step 6: Simplify the Fraction
To simplify:
\[ I = \frac{725760 \div 725760}{479001600 \div 725760} = \frac{1}{660} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{\frac{1}{660}} \]