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JEE Main20246 Apr 2024Morning ShiftMathematicsDefinite IntegrationActual

Let r_k= ₀^1 (1-x^7 )^k d x ₀^1 (1-x^7 )^ k+1 d x , k N . Then the value of _ k=1 ¹⁰ 1 7 (r_k-1 ) is equal to________

Correct answer

0

Step-by-step solution

aligned & I_K= 1 (1-x^7 )^K d x & I_K= . (1-x^7 )^K x |₀ ^1+7 K ₀^1 (1-x^7 )^ K-1 x^6 x d x & I_K=-7 K ₀^1 (1-x^7 )^ K-1 ( (1-x^7 )-1 ) d x & I_K=-7 K I_K+7 K I_ K-1 & I_K I_ K+1 = 7 K+8 7 K+7 aligned aligned & r_K= 7 K+8 7 ~K +7 & r _ K -1= 1 7( ~K +1) & 7 ( r _ K -1 )= 1 ~K +1 & _ K =1 ¹⁰( ~K +1)=11(6)-1=65 aligned

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