Question
The value of $$\int\limits_1^a {\left[ x \right]f'\left( x \right)} dx,\,a > 1$$ where $$\left[ x \right]$$ denotes the greatest integer not exceeding $$x$$ is-
A.
$$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .....f\left( {\left[ a \right]} \right)} \right\}$$
B.
$$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .....f\left( {\left[ a \right]} \right)} \right\}$$
C.
$$\left[ a \right]f\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .....f\left( a \right)} \right\}$$
D.
$$af\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .....f\left( a \right)} \right\}$$
Answer :
$$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .....f\left( {\left[ a \right]} \right)} \right\}$$
Solution :
Let $$a=k+ h$$ where $$k$$ is an integer such that $$\left[ a \right] = k$$ and $$0 \leqslant h < 1$$
$$\eqalign{
& \therefore \int\limits_1^a {\left[ x \right]f'\left( x \right)dx} = \int\limits_1^2 {1\,f'\left( x \right)dx} + \int\limits_2^3 {2\,f'\left( x \right)dx} + .....\int\limits_{k - 1}^k {\left( {k - 1} \right)dx} + \int\limits_k^{k + h} {k\,f'\left( x \right)dx} \cr
& = \left\{ {f\left( 2 \right) - f\left( 1 \right)} \right\} + 2\left\{ {f\left( 3 \right) - f\left( 2 \right)} \right\} + 3\left\{ {f\left( 4 \right) - f\left( 3 \right)} \right\} + ..... + \left( {k - 1} \right)\left\{ {f\left( k \right) - f\left( {k - 1} \right)} \right\} + k\left\{ {f\left( {k + h} \right) - f\left( k \right)} \right\} \cr
& = - f\left( 1 \right) - f\left( 2 \right) - f\left( 3 \right)..... - f\left( k \right) - k\,f\left( {k + h} \right) \cr
& = \left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + f\left( 3 \right) + .....f\left( {\left[ a \right]} \right)} \right\} \cr} $$