Math miscellaneousHard
Question
The value of 
[x]f′(x)dx, a > 1, where [x] denotes the greatest integer not exceeding x is
[x]f′(x)dx, a > 1, where [x] denotes the greatest integer not exceeding x isOptions
A.af(a) - {f(1) + f(2) + ... + f([a])}
B.[a] f(a) - {f(1) + f(2) + ... + f([a])}
C.[a] f([a]) - {f(1) + f(2) + ... + f(a)}
D.af([a]) - {f(1) + f(2) + ... + f(a)}
Solution
Let a = k + h, where [a] = k and 0 ≤ h < 1
∴[x]f′(x) dx = 
1f′(x)dx + 
2f′(x)dx + ..... 
(k - 1)dx + 
kf′(x) dx
{f(2) - f(1)} + 2{f(3) - f(2)} + 3{f(4) - f(3)}+.....+ (k-1) - {f(k) - f(k - 1)}+ k{f(k + h) - f(k)}
= - f(1) - f(2) - f(3) .... - f(k) + k f(k + h)
= [a] f(a) - {f(1) + f(2) + f(3) + ... + f([a])}
∴
[x]f′(x) dx = 1f′(x)dx + 2f′(x)dx + ..... (k - 1)dx + kf′(x) dx {f(2) - f(1)} + 2{f(3) - f(2)} + 3{f(4) - f(3)}+.....+ (k-1) - {f(k) - f(k - 1)}+ k{f(k + h) - f(k)}
= - f(1) - f(2) - f(3) .... - f(k) + k f(k + h)
= [a] f(a) - {f(1) + f(2) + f(3) + ... + f([a])}
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