How to integrate ? int_0^oo 1/(1+e^x)*dx∫∞011+ex⋅dx

1 Answer
mason m
May 6, 2017

ln(2)ln(2)

Explanation:

First working with the indefinite integral, divide the numerator and denominator by e^xex:

int1/(1+e^x)dx=inte^-x/(e^-x+1)dx∫11+exdx=∫e−xe−x+1dx

Let u=e^-x+1u=e−x+1, implying that du=-e^-xdxdu=−e−xdx:

=-int(-e^-x)/(e^-x+1)dx=-int1/udu=-lnabsu=−∫−e−xe−x+1dx=−∫1udu=−ln|u|

=-ln(e^-x+1)+C=−ln(e−x+1)+C

So:

int_0^oo1/(1+e^x)dx∫∞011+exdx

=(lim_(xrarroo)(-ln(e^-x+1)))-(-ln(e^0+1))

Note that lim_(xrarroo)e^-x=0:

=-ln(1)+ln(1+1)

=ln(2)

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