Question

The value of ${\int }_1^a\left[a\right]f^{\prime }\left(x\right)dx$, $a>1$, where [x] denotes the greatest integer not exceeding $x$ is

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

Answer 1

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

Let $I={\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx$, $a>1$
Let $a=k+h$ where $\left[a\right]=k$, $0\le h<1$
$\because$ ${\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx={\int }_1^{2}1f^{\prime }\left(x\right)dx+{\int }_2^{3}2f^{\prime }\left(x\right)dx+...+{\int }_{k-1}^k(k-1)f^{\prime }\left(x\right)dx+{\int }_k^{k+h}h{f}^{\prime }\left(x\right)dx$
$=(f\left(2\right)-f\left(1\right))+2(f\left(3\right)-f\left(2\right))+...+(k-1)[f\left(k\right)-f(k-1)]+k(f(k+h)-f\left(k\right))$
$=-f\left(1\right)-f\left(2\right)-f\left(3\right)...-f\left(k\right)+hf(k+h)$
$=\left[a\right]f\left(a\right)-\{f\left(1\right)+f\left(2\right)+...+f\left(\left[a\right]\right)\}$