Question

\( \int_{0}^{1} x(1 - x)^{99} \, dx = \)

• A. \( \frac{1}{10010} \)
• B. \( \frac{1}{1010} \)
• C. \( \frac{1}{10100} \)
• D. \( \frac{1}{10100} \)

Hint:

Beta function integrals of the form \( \int_0^1 x^m(1-x)^n dx \) directly reduce to factorial formNo need for lengthy integration.

Answer 1

The Correct Option is D

Step 1: Identify the standard form.
\[ \int_0^1 x^m (1-x)^n dx = \frac{m!n!}{(m+n+1)!} \]
Here:
\[ m = 1,\quad n = 99 \]

Step 2: Apply the formula.
\[ \int_0^1 x(1-x)^{99}dx = \frac{1!\cdot 99!}{101!} \]

Step 3: Simplify factorial expression.
\[ 101! = 101 \cdot 100 \cdot 99! \]
So:
\[ \frac{1!\cdot 99!}{101!} = \frac{99!}{101 \cdot 100 \cdot 99!} \]

Step 4: Cancel common terms.
\[ = \frac{1}{101 \cdot 100} \]

Step 5: Final simplification.
\[ = \frac{1}{10100} \]

Step 6: Verify with options.
Matches option (D).

Step 7: Final conclusion.
\[ \boxed{\frac{1}{10100}} \]