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

Step 1: Understanding the Concept:
Expanding \( (1-x)^{12} \) would be tedious. Instead, we use the definite integral property \( \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \).
Step 2: Detailed Explanation:
Applying the property to the given integral: \[ I = \int_{0}^{1} (1-x) [1 - (1-x)]^{12} dx \] \[ I = \int_{0}^{1} (1-x) x^{12} dx \] \[ I = \int_{0}^{1} (x^{12} - x^{13}) dx \] Integrating term by term: \[ I = \left[ \frac{x^{13}}{13} - \frac{x^{14}}{14} \right]_{0}^{1} \] \[ I = \left( \frac{1}{13} - \frac{1}{14} \right) - (0 - 0) \] Finding a common denominator: \[ I = \frac{14 - 13}{13 \times 14} = \frac{1}{182} \].
Step 3: Final Answer:
The integral equals 1/182.