Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 77687 by jagoll last updated on 09/Jan/20

given function f(x) =f(x+2) for ∀x∈R if ∫_0 ^2 f(x)dx=B , what value of ∫_3 ^7 f(x+8)dx = ?

$${given}\:{function} \\ $$$${f}\left({x}\right)\:={f}\left({x}+\mathrm{2}\right)\:{for}\:\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}={B}\:,\:{what}\:{value}\:{of} \\ $$$$\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{8}\right){dx}\:=\:? \\ $$

Commented by mr W last updated on 09/Jan/20

∫_3 ^7 f(x+8)dx =∫_3 ^7 f(x+4×2)dx =∫_3 ^7 f(x)dx =∫_3 ^7 f(x−3)d(x−3) =∫_0 ^4 f(x)dx =∫_0 ^2 f(x)dx+∫_2 ^4 f(x)dx =∫_0 ^2 f(x)dx+∫_2 ^4 f(x−2)d(x−2) =∫_0 ^2 f(x)dx+∫_0 ^2 f(x)dx =2∫_0 ^2 f(x)dx ⇒2B generally if f(x)=f(x+T) and ∫_0 ^T f(x)dx=B, ∫_a ^(a+nT) f(x+mT)dx=n∫_0 ^T f(x)dx=nB with n,m=integer

$$\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{8}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{4}×\mathrm{2}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}−\mathrm{3}\right){d}\left({x}−\mathrm{3}\right) \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}−\mathrm{2}\right){d}\left({x}−\mathrm{2}\right) \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$\Rightarrow\mathrm{2}{B} \\ $$$$ \\ $$$${generally} \\ $$$${if}\:{f}\left({x}\right)={f}\left({x}+{T}\right)\:{and}\:\underset{\mathrm{0}} {\overset{{T}} {\int}}\:{f}\left({x}\right){dx}={B}, \\ $$$$\underset{{a}} {\overset{{a}+{nT}} {\int}}\:{f}\left({x}+{mT}\right){dx}={n}\underset{\mathrm{0}} {\overset{{T}} {\int}}\:{f}\left({x}\right){dx}={nB} \\ $$$${with}\:{n},{m}={integer} \\ $$

Commented by jagoll last updated on 09/Jan/20

thanks you sir

$${thanks}\:{you}\:{sir} \\ $$

Commented by jagoll last updated on 09/Jan/20

sir W how about f(x)= f(x+7)∀x∈R if ∫_2 ^4 f(x)dx=5 and ∫_3 ^4 f(x)dx=2 what is ∫_9 ^(10) f(x)dx=?

$${sir}\:{W}\:{how}\:{about}\:{f}\left({x}\right)=\:{f}\left({x}+\mathrm{7}\right)\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right){dx}=\mathrm{5}\:{and}\:\underset{\mathrm{3}} {\overset{\mathrm{4}} {\int}}{f}\left({x}\right){dx}=\mathrm{2}\: \\ $$$${what}\:{is}\:\underset{\mathrm{9}} {\overset{\mathrm{10}} {\int}}{f}\left({x}\right){dx}=? \\ $$

Commented by mr W last updated on 09/Jan/20

∫_9 ^(10) f(x)dx =∫_9 ^(10) f(x−7)d(x−7) =∫_2 ^3 f(x)dx =∫_2 ^4 f(x)dx−∫_3 ^4 f(x)dx =5−2 =3

$$\underset{\mathrm{9}} {\overset{\mathrm{10}} {\int}}{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{9}} {\overset{\mathrm{10}} {\int}}{f}\left({x}−\mathrm{7}\right){d}\left({x}−\mathrm{7}\right) \\ $$$$=\underset{\mathrm{2}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}{f}\left({x}\right){dx}−\underset{\mathrm{3}} {\overset{\mathrm{4}} {\int}}{f}\left({x}\right){dx} \\ $$$$=\mathrm{5}−\mathrm{2} \\ $$$$=\mathrm{3} \\ $$

Commented by jagoll last updated on 09/Jan/20

thanks you sir

$${thanks}\:{you}\:{sir} \\ $$

Commented by jagoll last updated on 09/Jan/20

sir line 8 is ∫_0 ^2 f(x)dx+∫_0 ^2 f(x)dx . is correct sir?

$${sir}\:{line}\:\mathrm{8}\:{is}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}+\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}\:.\:{is}\:{correct}\:{sir}? \\ $$

Commented by jagoll last updated on 09/Jan/20

i think it your typo sir

$${i}\:{think}\:{it}\:{your}\:{typo}\:{sir} \\ $$

Commented by mr W last updated on 09/Jan/20

yes, now fixed!

$${yes},\:{now}\:{fixed}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com