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What is #int_-oo^oo (e^x)/((e^(2x))+3) dx#?

Answer 1

Emilia Aguilar

#pi/sqrt3#

#int_-oo^oo e^x/(e^(2x)+3)dx=#

Using trigonometric substitution #e^x=sqrt3tany# #e^x*dx=sqrt3sec^2y*dy#

Since #tany=e^x/sqrt3#, when #x->-oo# => #tany -> -oo# => #y->(-pi/2)^"+"# #x->oo# => #tany -> oo# => #y->(pi/2)^"-"#

So the expression becomes

#=int_((-pi/2)^"+")^((pi/2)^"-") (sqrt3sec^2y)/(3tan^2y+3)dy=int_((-pi/2)^"+")^((pi/2)^"-") (sqrt3cancel(sec^2y))/(3cancel(sec^2y))dy# #=(y/sqrt3)|_((-pi/2)^"+")^((pi/2)^"-")# #=1/sqrt3[pi/2-(-pi/2)]=pi/sqrt3#

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Answer 2

Aurora Alvarado

The integral of (e^x)/((e^(2x))+3) with respect to x from negative infinity to positive infinity is equal to 1/3.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.