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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

Let f be a differentiable function defined in [0,1] such that f(f(x))=x and f(0)=1 . If the value of ₀^1(x-f(x))²⁰¹⁸ d x= p q where p and q are relatively prime then find the value of (p+q) .

Correct answer

2020

Step-by-step solution

I= ₀^1(x-f(x))²⁰¹⁸ d x ...(1) Putting, aligned & x=f(t) d x=f^ (t) d t & I= ₁^0(f(t)-f(f(t)))²⁰¹⁸ f^ (t) d t f(f(0))=0 f(1)=0 & I=- ₀^1(f(t)-t)²⁰¹⁸ f^ (t) d t & I=- ₀^1(t-f(t))²⁰¹⁸ f^ (t) d t .....(2) aligned Eqn. (1) + Eqn. (2) aligned 2 I & = ₀^1(x-f(x))²⁰¹⁸ (1-f^ (x) ) d x= ( (x-f(x))²⁰¹⁹ 2019 )₀^1 & = 1 2019 - ( -1 2019 ) I & = 1 2019 p q (p+q)=2020 aligned

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