Question

The value of ${\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx,a>1$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is

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

Answer 1

The correct option is D $\left[a\right]f\left(a\right)-\left(f\right(1)+f(2)+\dots \dots +f(a\left)\right)$

${\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx$
$={\int }_1^{2}1f^{\prime }\left(x\right)dx+{\int }_2^{3}2f^{\prime }\left(x\right)dx+{\int }_3^{4}3f^{\prime }\left(x\right)dx+....+{\int }_{a-1}^a[a-1]f^{\prime }\left(x\right)dx$
$=[{f\left(x\right)]}_1^2+2[{f\left(x\right)]}_2^3+3[{f\left(x\right)]}_3^4+......[a-1\left]\right[{f\left(x\right)]}_{a-1}^a$
$=f\left(2\right)-f\left(1\right)+2f\left(3\right)-2f\left(2\right)+3f\left(4\right)-3f\left(3\right)+.....+(a-1)f\left(a\right)-[a-1]f(a-1)$
$I=[a-1]f\left(a\right)-\left\{f\right(1)+f(2)+f(3).....+f(a-1\left)\right\}$
$\Rightarrow I=\left[a\right]f\left(a\right)-\left\{f\right(1)+f(2)+f(3).....+f(\left[a\right]\left)\right\}$ ($\because \left[a\right]=a-1,whena>1$