Find #int_0^1f(x)dx# ?

Question

#f:RR->RR#, continuous in #RR# with #f(x)e^x+f(e^(-x))=xe^x+e^(-x)#
, #AA##x##in##RR#

Find #int_0^1f(x)dx#

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Answer 1 · 2018-05-30T12:28:27

Shown below:

By inspection, #f(x) = x #

#=> int_0 ^1 x dx #

#int x^n dx = 1/(n+1) x^(n+1) + c #

#=> int_0 ^1 x^1 dx = [1/2 x^2 ]_0 ^1 #

#=> ( 1/2 * 1^2 ) - ( 1/2 * 0^2 ) #

# = 1/2 #