Question

\(\int_0^1 x(1-x)^{12 dx\) is equal to}

• A. 1/132
• B. 1/156
• C. 1/182
• D. None of these

Hint:

Always expand factorial properly: \((n+2)! = (n+2)(n+1)n!\).

Answer 1

The Correct Option is C

Concept: \[ \int_0^1 x^m (1-x)^n dx = \frac{m!n!}{(m+n+1)!} \]

Step 1: Apply formula.
\[ m=1,\; n=12 \] \[ = \frac{1!\cdot12!}{14!} \]

Step 2: Simplify factorials.
\[ 14! = 14 \cdot 13 \cdot 12! \] \[ \Rightarrow \frac{12!}{14!} = \frac{1}{14 \cdot 13} \] \[ = \frac{1}{182} \]