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JEE Main2003MathematicsDefinite IntegrationActual

The value of the integral I= ₀^1 x(1-x)^n d x is

Options

  1. A1 n+1 + 1 n+2
  2. B1 n+1
  3. C1 n+2
  4. D1 n+1 - 1 n+2

Correct answer

D. 1 n+1 - 1 n+2

Step-by-step solution

I= ₀^1 x(1-x)^n d x-I= ₀^1-x(1-x)^n d x= ₀^1(1-x-1)(1-x)^n d x= ₀^1(1-x)^ n+1 d x- ₀^1(1-x)^n d x= [ (1-x)^ n+2 -(n+2) ]₀^1- [ (1-x)^ n+1 -(n+1) ]₀^1= 1 n+2 - 1 n+1 I= 1 n+1 - 1 n+2

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