a > 1, ; _1^a [ x ]f' ( x )dx = - Vidyadip

MCQ

$a > 1,\;\mathop \smallint \limits_1^a \left[ x \right]f'\left( x \right)dx = $

A
$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .\;.\;.\; + f\left( {\left[ a \right]} \right)} \right\}$
✓
$\;\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .\;.\;.\; + f\left( {\left[ a \right]} \right)} \right\}$
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

Correct option: B.

$\;\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .\;.\;.\; + f\left( {\left[ a \right]} \right)} \right\}$

b
$a=k+h$ where $k$ is an integer such that

$[a]=k$ and $0 \leq h<1$

$\therefore \int_{1}^{a}[x] f^{\prime}(x) d x=\int_{1}^{2} 1 f^{\prime}(x) d x \quad+\int_{2}^{3} 2 f^{\prime}(x) d x+\ldots . $$\int_{k-1}^{k}(k-1) d x+\int_{k}^{k+h} k f^{\prime}(x) d x$

$\{f(2)-f(2)\}+2\{f(3)-f(2)\}+3\{f(4)-f(3)\}+\ldots $$\ldots+(k-1)\{f(k)-f(k-1)+k\{f(k+h)-f(k)\}$

$=-f(1)-f(2)-f(3) \ldots \ldots-f(k)+k f(k+h)$

$=[a] f(a)-\{f(1)+f(2)+f(3)+\ldots \ldots \ldots f([a])\}$

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