Let y=f(x) be a differentiable function such that f(x)= ₀^x ( f(t) t + t ) d t x0 and f(1)=0 . If L= _ x 1 f(x) ^2 x , then find the value of [ 1 L ] . [Note: Where [ ] denotes greatest integer function.]

Question

Let y=f(x) be a differentiable function such that f(x)= ₀^x ( f(t) t + t ) d t x>0 and f(1)=0 . If L= _ x 1 f(x) ^2 x , then find the value of [ 1 L ] . [Note: Where [ ] denotes greatest integer function.]

Quantrex Academy · Accepted Answer

Correct answer: 6

aligned f^ (x) & = f(x) x + x f(x) 1 x = x x d x+C f(x) x & = ( x)^2 2 +C f(x)= x( x)^2 2 L & = _ x 1 f(x) ^2 x = _ x 1 ( (1+(x-1)))^2 2 ^2(x-1)^2 = 1 2 ^2 [ 1 L ] & =2 aligned

Let y=f(x) be a differentiable function such that f(x)= ₀^x ( f(t) t + t ) d t x>0 and f(1)=0 . If L= _ x 1 f(x) ^2 x , then find the value of [ 1 L ] . [Note: Where [ ] denotes greatest integer function.]

Correct answer

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