Question #3983

Integrate e[sup]x[/sup]/( √(e[sup]x[/sup]+1))dx & & & from 0 to ln3

Expert's answer

Int{e^x/sqrt(1+e^x)}dx

substitution& sqrt(e^x+1)=t& =>& e^x=t^2-1& =>& x=ln(t^2-1)& =>& dx=2tdt/(t^2-1)

limits for new variable: x=0 => t=sqrt(2)

x=ln3 => t=2

so we have

Int{(t^2-1)/t*2t/(t^2-1)}dt=Int{2}dt=t

Using Newton-Leibniz formula for limits sqrt(2),2 we obtain

2(2-sqrt(2))=4-2sqrt(2) and that's it

substitution& sqrt(e^x+1)=t& =>& e^x=t^2-1& =>& x=ln(t^2-1)& =>& dx=2tdt/(t^2-1)

limits for new variable: x=0 => t=sqrt(2)

x=ln3 => t=2

so we have

Int{(t^2-1)/t*2t/(t^2-1)}dt=Int{2}dt=t

Using Newton-Leibniz formula for limits sqrt(2),2 we obtain

2(2-sqrt(2))=4-2sqrt(2) and that's it

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