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The value of ∫₁^a [x] f'(x) dx, a > 1 where [x] denotes the greatest integer not exceeding x is

1. a f(a) - f(1) + f(2) +..........f([a])
2. [a]f(a) - f(1) + f(2) +..........f([a])
3. [a]f([a]) - f(1) + f(2) +..........f(a)
4. a f([a]) - f(1) + f(2) +..........f(a)

Correct answer: [a]f(a) - f(1) + f(2) +..........f([a])

Solution

Split the integral over [1,2],[2,3],...,[[a],a] using [x]=k on (k,k+1): sum_k k(f(k+1)-f(k)) + [a](f(a)-f([a])). Collecting terms gives [a]f(a) - (f(1)+f(2)+...+f([a])).

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