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JEE MainMathematicsDefinite Integration

Let I(m,n) = ₀^1 x^m (1-x)^n dx for positive integers m and n . The value of the ratio I(5, 9) I(6, 8) is

Options

  1. A2 3
  2. B5 8
  3. C3 2
  4. D4 3

Correct answer

C. 3 2

Step-by-step solution

We are given I(m,n) = ₀^1 x^m (1-x)^n dx . Consider the integral for I(6,8) : I(6,8) = ₀^1 x^6 (1-x)^8 dx Applying integration by parts, taking x^6 as the first function and (1-x)^8 as the second function: I(6,8) = [ x^6 -(1-x)^9 9 ]₀^1 - ₀^1 6x^5 ( -(1-x)^9 9 ) dx The first term evaluates to zero at both the upper limit x=1 and the lower limit x=0 . I(6,8) = 6 9 ₀^1 x^5 (1-x)^9 dx I(6,8) = 2 3 I(5,9) Rearranging the terms, we get: I(5,9) I(6,8) = 3 2 Answer: 3 2

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