The value of `int_1^a[x]f^(prime)(x)dx `,where `a >1`, and `[x]` denotes the greatest integer not exceeding x, is
(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)+.......+fA}`
(D) `af([a])-{f(1)+f(2)+......+fA}`

Options

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))`

Answer

B

Explanation

Let`=int_(1)^(a)[x]f'(x)dx, agt1`
Let `a=k+h`, where `[a]=k` and `0le hlt1`
`:. int_(1)^(a)[x]f'(x)dx=int_(1)^(2)1 f'(x)dx+int_(2)^(3)2f'(x)dx`
`+……………+int_(k-1)^(k)f'(x)dx+int_(k)^(k+h)kf'(x)dx`
`=[f(2)-f(1)]+2[f(3)-f(2)]+……….`
`+(k-1)[f(k)-f(k-1)]+k[f(k+h)-f(k)]`
`=-f(1)-f(2)-f(3)-.............f(k)+kf(k+h)`
`=[a]f(a)-[f(1)+f(2)+......+f([a])]`