The value of `int_1^a[x]f^(prime)(x)dxf^(prime)(x)dx ,w h e r ea >1,a n d[x]` denotes the greatest integer not exceeding `x ,` is `af(a)-{f(1)f(2)++f([a])}` `[a]f(a)-{f(1)+f(2)++f([a])}` `[a]f(a)-{f(1)+f(2)++fA}` `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])]`