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Question Number 31145 by Joel578 last updated on 03/Mar/18

Given  ∫_0 ^1  f(x) dx =  (((2018)),((    0)) ) + (1/2) (((2018)),((    1)) ) + (1/3) (((2018)),((    2)) ) + ... + (1/(2019)) (((2018)),((2018)) )  ∫_0 ^1  g(x) dx =  (((2018)),((    0)) ) − (1/2) (((2018)),((    1)) ) + (1/3) (((2018)),((    2)) ) − ... + (1/(2019)) (((2018)),((2018)) )  h(x) is an odd function  Then what is the value of ∫_(−3) ^( 3)  f(x).g(x).h(x) dx ?

$$\mathrm{Given} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{g}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:−\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$${h}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function} \\ $$$$\mathrm{Then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{−\mathrm{3}} ^{\:\mathrm{3}} \:{f}\left({x}\right).{g}\left({x}\right).{h}\left({x}\right)\:{dx}\:? \\ $$

Answered by ajfour last updated on 03/Mar/18

f(x)=(1+x)^(2018)     ,  g(x)=(1−x)^(2018)   ⇒  f(x)g(x)=(1−x^2 )^(2018)    (even function)  so f(x)g(x)h(x)   is  odd  ⇒  ∫_(−3) ^(  3) f(x)g(x)h(x)dx =0  .

$${f}\left({x}\right)=\left(\mathrm{1}+{x}\right)^{\mathrm{2018}} \:\:\:\:,\:\:{g}\left({x}\right)=\left(\mathrm{1}−{x}\right)^{\mathrm{2018}} \\ $$$$\Rightarrow\:\:{f}\left({x}\right){g}\left({x}\right)=\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2018}} \:\:\:\left({even}\:{function}\right) \\ $$$${so}\:{f}\left({x}\right){g}\left({x}\right){h}\left({x}\right)\:\:\:{is}\:\:{odd} \\ $$$$\Rightarrow\:\:\int_{−\mathrm{3}} ^{\:\:\mathrm{3}} {f}\left({x}\right){g}\left({x}\right){h}\left({x}\right){dx}\:=\mathrm{0}\:\:. \\ $$

Commented by Joel578 last updated on 04/Mar/18

Thank you very much

$${Thank}\:{you}\:{very}\:{much} \\ $$

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