The Student Room Group

this integral

could someone kind of walk me through this integral?

∫(1/(e^(2x)*(1+e^(-x)))dx

that's (in words) the integral (respect to x) of 1 over all this: e^(2x) times (1+e^(-x))

i think the easiest way to start is to bring the e^(2x) to the numerator by making it a e^(-2x).....but then what?



thanks so much guys
Reply 1
I = ∫ 1/[e^(2x) (1+ e^(-x))] dx
= ∫ e^(-2x)/[1 + e^(-x)] dx

Let u=e^(-x), then du=-e^(-x)dx. So the integral becomes:
I = ∫ -u/(1+u) du
= - ∫ (u+1-1)/(u+1) du
= - ∫ (u+1)/(u+1) du + ∫ 1/(u+1) du
= -u + ln|1+u| + C
= ln[1+e^(-x)] - e^(-x) + C

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