Question

The value of $\underset{1}{\overset{a}{\int }}\left[x\right]f^{\prime }\left(x\right)dx,a>1,$ where $\left[x\right]$ denotes the greatest integer not exceeding $x$ is

• A. $af\left(a\right)-\left\{f\right(1)+f(2)+....f(\left[a\right]\left)\right\}$
• B. $\left[a\right]f\left(a\right)-\left\{f\right(1)+f(2)+....f(\left[a\right]\left)\right\}$
• C. $\left[a\right]f\left(\right[a\left]\right)-\left\{f\right(1)+f(2)+......+f(a\left)\right\}$
• D. $af\left(\right[a\left]\right)-\left\{f\right(1)+f(2)+.....+f(a\left)\right\}$

Answer 1

The correct option is B $\left[a\right]f\left(a\right)-\left\{f\right(1)+f(2)+....f(\left[a\right]\left)\right\}$

Let $I=\underset{1}{\overset{a}{\int }}\left[x\right]f^{\prime }\left(x\right)dx,a>1$
Let $a=k+h,$ where $\left[a\right]=k,$ and $0\le h<1$
$\therefore {\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx$
$=\underset{1}{\overset{2}{\int }}1f^{\prime }\left(x\right)dx+\underset{2}{\overset{3}{\int }}2f^{\prime }\left(x\right)dx+.....+$
$\underset{k-1}{\overset{k}{\int }}(k-1)f^{\prime }\left(x\right)dx+\underset{k}{\overset{k+h}{\int }}k{f}^{\prime }\left(x\right)dx$
$=\left[f\right(2)-f(1\left)\right]+2\left[f\right(3)-f(2\left)\right]+.....+(k-1)\left[f\right(k)-f(k-1\left)\right]+k\left[f\right(k+h)-f(k\left)\right]$
$=-f\left(1\right)-f\left(2\right)-f\left(3\right).....-f\left(k\right)+kf(k+h)$
$=\left[a\right]f\left(a\right)-\left[f\right(1)+f(2)+....f(\left[a\right]\left)\right]$