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2-6-x-1-x-2-x-3-x-9-dx-a-1-b-0-c-6-d-2-e-4-




Question Number 150247 by mathdanisur last updated on 10/Aug/21
∫_( 2) ^( 6)  (x-1)(x-2)(x-3)...(x-9) dx = ?  a)1     b)0     c)6!     d)-2    e)4!
$$\underset{\:\mathrm{2}} {\overset{\:\mathrm{6}} {\int}}\:\left(\mathrm{x}-\mathrm{1}\right)\left(\mathrm{x}-\mathrm{2}\right)\left(\mathrm{x}-\mathrm{3}\right)…\left(\mathrm{x}-\mathrm{9}\right)\:\mathrm{dx}\:=\:? \\ $$$$\left.\mathrm{a}\left.\right)\left.\mathrm{1}\left.\:\left.\:\:\:\:\mathrm{b}\right)\mathrm{0}\:\:\:\:\:\mathrm{c}\right)\mathrm{6}!\:\:\:\:\:\mathrm{d}\right)-\mathrm{2}\:\:\:\:\mathrm{e}\right)\mathrm{4}! \\ $$
Commented by amin96 last updated on 10/Aug/21
∫_2 ^6 Π_(n=1) ^9 (x−n)dx=Π_(n=1) ^9 ∫_2 ^6 (x−n)dx=  =Π_(n=1) ^9 (∣_2 ^6 ((x^2 /2)−nx))=Π_(n=1) ^9 (16−4n)=12∙8∙4∙0....=0
$$\int_{\mathrm{2}} ^{\mathrm{6}} \underset{{n}=\mathrm{1}} {\overset{\mathrm{9}} {\prod}}\left({x}−{n}\right){dx}=\underset{{n}=\mathrm{1}} {\overset{\mathrm{9}} {\prod}}\int_{\mathrm{2}} ^{\mathrm{6}} \left({x}−{n}\right){dx}= \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{9}} {\prod}}\left(\mid_{\mathrm{2}} ^{\mathrm{6}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−{nx}\right)\right)=\underset{{n}=\mathrm{1}} {\overset{\mathrm{9}} {\prod}}\left(\mathrm{16}−\mathrm{4}{n}\right)=\mathrm{12}\centerdot\mathrm{8}\centerdot\mathrm{4}\centerdot\mathrm{0}….=\mathrm{0} \\ $$
Commented by mr W last updated on 10/Aug/21
maybe the question is  ∫_( 2) ^( 8)  (x-1)(x-2)(x-3)...(x-9) dx = ?
$${maybe}\:{the}\:{question}\:{is} \\ $$$$\underset{\:\mathrm{2}} {\overset{\:\mathrm{8}} {\int}}\:\left(\mathrm{x}-\mathrm{1}\right)\left(\mathrm{x}-\mathrm{2}\right)\left(\mathrm{x}-\mathrm{3}\right)…\left(\mathrm{x}-\mathrm{9}\right)\:\mathrm{dx}\:=\:? \\ $$
Commented by amin96 last updated on 10/Aug/21
Thanks)
$$\left.{Thanks}\right) \\ $$
Commented by mathdanisur last updated on 10/Aug/21
Thank you ser cool
$${Thank}\:{you}\:{ser}\:{cool} \\ $$
Commented by mr W last updated on 10/Aug/21
obviously wrong!  ∫_a ^b f(x)g(x)dx≠(∫_a ^b f(x)dx)×(∫_a ^b g(x)dx)
$${obviously}\:{wrong}! \\ $$$$\int_{{a}} ^{{b}} {f}\left({x}\right){g}\left({x}\right){dx}\neq\left(\int_{{a}} ^{{b}} {f}\left({x}\right){dx}\right)×\left(\int_{{a}} ^{{b}} {g}\left({x}\right){dx}\right) \\ $$
Commented by mr W last updated on 10/Aug/21
all answers given are wrong!
$${all}\:{answers}\:{given}\:{are}\:{wrong}! \\ $$
Commented by mr W last updated on 10/Aug/21
just expand and you′ll get the right  answer −((1664)/5).
$${just}\:{expand}\:{and}\:{you}'{ll}\:{get}\:{the}\:{right} \\ $$$${answer}\:−\frac{\mathrm{1664}}{\mathrm{5}}. \\ $$
Commented by mathdanisur last updated on 10/Aug/21
Thankyou Ser, so what′s the solution now.?
$$\mathrm{Thankyou}\:\mathrm{Ser},\:\mathrm{so}\:\mathrm{what}'\mathrm{s}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{now}.? \\ $$
Commented by mr W last updated on 10/Aug/21
∫_( 2) ^( 8)  (x-1)(x-2)(x-3)...(x-9) dx = 0
$$\underset{\:\mathrm{2}} {\overset{\:\mathrm{8}} {\int}}\:\left(\mathrm{x}-\mathrm{1}\right)\left(\mathrm{x}-\mathrm{2}\right)\left(\mathrm{x}-\mathrm{3}\right)…\left(\mathrm{x}-\mathrm{9}\right)\:\mathrm{dx}\:=\:\mathrm{0} \\ $$
Commented by mr W last updated on 10/Aug/21
∫_( 5−a) ^( 5+a)  (x-1)(x-2)(x-3)...(x-9) dx = 0  a∈R
$$\underset{\:\mathrm{5}−{a}} {\overset{\:\mathrm{5}+{a}} {\int}}\:\left(\mathrm{x}-\mathrm{1}\right)\left(\mathrm{x}-\mathrm{2}\right)\left(\mathrm{x}-\mathrm{3}\right)…\left(\mathrm{x}-\mathrm{9}\right)\:\mathrm{dx}\:=\:\mathrm{0} \\ $$$${a}\in\mathbb{R} \\ $$
Commented by mathdanisur last updated on 10/Aug/21
Thank You Ser
$$\mathrm{Thank}\:\mathrm{You}\:\boldsymbol{\mathrm{S}}\mathrm{er} \\ $$

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