Question

The value of \( \int_0^1 x(1 - x)^{10} \, dx \) is equal to:

• A. \( \frac{1}{110} \)
• B. \( \frac{1}{132} \)
• C. \( \frac{1}{156} \)
• D. \( \frac{1}{90} \)
• E. \( \frac{5}{156} \)

Hint:

When dealing with integrals of the form \( x^a(1 - x)^b \), use the Beta function formula: \[ B(a+1, b+1) = \int_0^1 x^a (1 - x)^b \, dx. \] This simplifies the computation of such integrals.

Answer 1

The Correct Option is B

We evaluate the integral: \[ I = \int_0^1 x(1 - x)^{10} \, dx. \] Step 1: Using Beta Function The given integral resembles the Beta function: \[ B(m+1, n+1) = \int_0^1 x^m (1-x)^n \, dx = \frac{m! \, n!}{(m+n+1)!}. \] Comparing, we identify \( m = 1 \) and \( n = 10 \), so: \[ I = B(2, 11) = \frac{1! \cdot 10!}{(2+11-1)!} = \frac{1 \cdot 10!}{12!}. \] Step 2: Simplifying Factorials Expanding factorials: \[ I = \frac{10!}{12 \times 11 \times 10!} = \frac{1}{12 \times 11} = \frac{1}{132}. \]
Final Answer: (B) \( \frac{1}{132} \).