What is the value of the integral I = int fra | vedclass

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Question

(A) Given the integral $I = \int \frac{dx}{(1 + e^x)(1 + e^{-x})}$.Rewrite the term $(1 + e^{-x})$ as $(1 + \frac{1}{e^x}) = \frac{e^x + 1}{e^x}$.Subs

Vedclass · Accepted Answer

$\frac{-1}{1 + e^x}$

Vedclass · Answer

(A) Given the integral $I = \int \frac{dx}{(1 + e^x)(1 + e^{-x})}$.Rewrite the term $(1 + e^{-x})$ as $(1 + \frac{1}{e^x}) = \frac{e^x + 1}{e^x}$.Substituting this into the integral,we get $I = \int \frac{dx}{(1 + e^x) \cdot \frac{e^x + 1}{e^x}} = \int \frac{e^x dx}{(1 + e^x)^2}$.Let $t = 1 + e^x$,then $dt = e^x dx$.Substituting these into the integral,$I = \int \frac{dt}{t^2} = \int t^{-2} dt$.Integrating with respect to $t$,we get $I = \frac{t^{-1}}{-1} + C = -\frac{1}{t} + C$.Substituting back $t = 1 + e^x$,we get $I = -\frac{1}{1 + e^x} + C$.

What is the value of the integral $I = \int \frac{dx}{(1 + e^x)(1 + e^{-x})}$?

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