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Jonask
Group Title
\[\huge I= \int _0^a \frac{f(x)}{f(xa)+f(x)}\]
show that I=a/2
 one year ago
 one year ago
Jonask Group Title
\[\huge I= \int _0^a \frac{f(x)}{f(xa)+f(x)}\] show that I=a/2
 one year ago
 one year ago

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hartnn Group TitleBest ResponseYou've already chosen the best response.1
any more info about f(x) even function or odd function ?
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
basically you need to use this : \(\huge \int \limits _a^b f(x)dx=\int \limits_a^b f(a+bx)dx\)
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
\(\huge I= \int _0^a \frac{f(x)}{f(xa)+f(x)} \\\huge I =\int _0^a \frac{f(ax)}{f(x)+f(ax)} \)
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
if f(x)=f(x) then you get your result after adding the above 2 equations....
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
i tried a u=xa substitution
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
no info about even or odd
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
oh sry its ax the original quest\[\huge \int_0^a\frac{f(x)}{f(x)+f(ax)}dx\]
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
then using that property will directly give you answer.... and ofcourse you have to add 2 equations u got....
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
using\[\frac{ a}{a+b}=\frac{ b }{ a+b }+1\] so \[\int\limits_0^a \frac{ f(x) }{ f(x)+f(ax) }=\int\limits_0^a \frac{ f(ax) }{ f(x)+f(xa)}+1\]
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
is that required to be used ? you can use the property...
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
u=ax du=dx \[I=()\int_a^0\color{red}{\frac{f(u)}{f(u)+f(au)}}du+\int_0^a1\] red same as I so \[I_0^a+I_a^0=a\]
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
wich property?
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
\(\huge \int \limits _a^b f(x)dx=\int \limits_a^b f(a+bx)dx\)
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
why is this true
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.1
you want proof ? i need to look up...
 one year ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
no wait before proof let me see what it gives
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.1
I remember the proof.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.1
in RHS, let a+bx = t => dx = dt limits also change to x=b to a just substitute and you'll get the LHS
 one year ago
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