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Question Number 74068 by FCB last updated on 18/Nov/19

Commented by FCB last updated on 19/Nov/19

$thank you$
Commented by FCB last updated on 18/Nov/19

$[e^{x}] - Integerpart (whole part)$
Commented by mathmax by abdo last updated on 18/Nov/19
$changement e^x =t give ∫_0 ^2 [e^x ]dx = ∫_1 ^e^2 [t](dt/t) e^2 ∼7,389 ⇒∫_0 ^2 [e^x ]dx =∫_1 ^(7,389) (([t])/t)dt =Σ_(k=1) ^6 ∫_k ^(k+1) (([t])/t)dt + ∫_7 ^(7,389) (([t])/t)dt =Σ_(k=1) ^6 k {ln(k+1)−ln(k)} +7 {ln(e^2 )−ln(7)} =Σ_(k=1) ^6 kln(((k+1)/k)) +7{2−ln(7)} =ln(2)+2ln((3/2))+3ln((4/3))+4ln((5/4))+5ln((6/5))+6ln((7/6)) +7{2−ln(7)}.$
$c h a n g e m e n t e^{x} = t g i v e \int_{0}^{2} [e^{x}] d x = \int_{1}^{e^{2}} [t] \frac{d t}{t}$ $e^{2} \sim 7, 389 \Rightarrow \int_{0}^{2} [e^{x}] d x = \int_{1}^{7, 389} \frac{[t]}{t} d t$ $= \sum_{k = 1}^{6} \int_{k}^{k + 1} \frac{[t]}{t} d t + \int_{7}^{7, 389} \frac{[t]}{t} d t$ $= \sum_{k = 1}^{6} k {l n (k + 1) - l n (k)} + 7 {l n (e^{2}) - l n (7)}$ $= \sum_{k = 1}^{6} k l n (\frac{k + 1}{k}) + 7 {2 - l n (7)}$ $= l n (2) + 2 l n (\frac{3}{2}) + 3 l n (\frac{4}{3}) + 4 l n (\frac{5}{4}) + 5 l n (\frac{6}{5}) + 6 l n (\frac{7}{6})$ $+ 7 {2 - l n (7)} .$
Commented by abdomathmax last updated on 19/Nov/19

$y o u a r e w e l c o m e .$
	Answered by Tanmay chaudhury last updated on 18/Nov/19

	$e^{x} = 1 w h e n x = 0$ $e^{x} = 2 w h e n x = l n 2$ $e^{x} = 3 w h e n x = l n 3$ $e^{x} = 4 w h e n x = l n 4$ $e^{x} = 5 w h e n x = l n 5$ $e^{x} = 6 w h e n x = l n 6$ $e^{x} = 7 w h e n x = l n 7$ $e^{x} = 8 w h e n x = l n 8 \to x = 2.079 b e y o n d g i v e n i n t e r v a l$ $\int_{0}^{2} [e^{x}] d x$ $= \int_{0}^{l n 2} 1 \times d x + \int_{l n 2}^{l n 3} 2 d x + \int_{l n 3}^{l n 4} 3 d x + \int_{l n 4}^{l n 5} 4 d x + \int_{l n 5}^{l n 6} 5 d x + \int_{l n 6}^{l n 7} 6 d x =$ $l n 2 + 2 (l n 3 - l n 2) + 3 (l n 4 - l n 3) + 4 (l n 5 - l n 4) + 5 (l n 6 - l n 5) + 6 (l n 7 - l n 6)$ $= - l n 2 - l n 3 - l n 4 - l n 5 - l n 6 + 6 l n 7$ $= l n (\frac{7^{6}}{2 \times 3 \times 4 \times 5 \times 6}) = l n (\frac{7^{6}}{720}) = 6 l n 7 - l n 720$ $= 11.68 - 6.58 \approx 5.10 p l s c h e c k . . .$
	Commented by mr W last updated on 19/Nov/19

	$= \int_{0}^{l n 2} 1 \times d x + \int_{l n 2}^{l n 3} 2 d x + \int_{l n 3}^{l n 4} 3 d x + \int_{l n 4}^{l n 5} 4 d x + \int_{l n 5}^{l n 6} 5 d x + \int_{l n 6}^{l n 7} 6 d x + \int_{\ln 7}^{2} 7 d x$ $= l n 2 + 2 (l n 3 - l n 2) + 3 (l n 4 - l n 3) + 4 (l n 5 - l n 4) + 5 (l n 6 - l n 5) + 6 (l n 7 - l n 6) + 7 (2 - \ln 7)$ $= - l n 2 - l n 3 - l n 4 - l n 5 - l n 6 - l n 7 + 7 \times 2$ $= 2 \times 7 - \ln (2 \times 3 \times . . . \times 7)$ $= 14 - \ln (7!)$ $\approx 5.4748$
	Commented by FCB last updated on 18/Nov/19

	$thank you sir$
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