₀¹ x(1-x)^{99} d x=

Correct Answer is : 1101001/10100101001

Correct Answer is : 1101001/10100101001

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COMEDK 2025 Evening Solved Paper

Question : 135 of 180

Marks: +1, -0

\int_0^1 x(1-x)^{99} d x=

Solution:

Concept:

Use the Beta function to evaluate integrals of the form

\int_0^1 x^m (1-x)^n \, dx

Explanation:

The given integral is

\int_0^1 x (1-x)^{99} \, dx

Compare with the Beta integral:

\int_0^1 x^{a-1} (1-x)^{b-1} \, dx = B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}

Here, exponent of

x

1

, so

a-1 = 1

gives

a = 2

Exponent of

(1-x)

99

, so

b-1 = 99

gives

b = 100

Thus, the integral equals

B(2,100) = \frac{\Gamma(2)\Gamma(100)}{\Gamma(102)} = \frac{1! \cdot 99!}{101!}

Simplify:

\frac{99!}{101!} = \frac{1}{100 \cdot 101} = \frac{1}{10100}

Answer:

\frac{1}{10100}

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