JEE Main20248 Apr 2024Morning ShiftMathematicsDefinite IntegrationActual

The value of k N for which the integral I_n= ₀^1 (1-x^k )^n d x, n N , satisfies 147 I₂₀=148 I₂₁ is

Options

Correct answer

D. 7

Step-by-step solution

aligned & I_n= ₀^1 (1-x^k )^n 1 d x & I_n= (1-x^k )^n x-n k ₀^1 (1-x^k )^ n-1 x^ k-1 d x & I_n=n k ₀^1 [ (1-x^k )^n- (1-x^k )^ n-1 ] d x & I_n=n k I_n-n k I_n & I_n I_ n-1 = n k n k+1 & I₂₁ I₂₀ = 21 k 1+21 k & = 147 148 k=7 aligned

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