Integrate the function e^ x (1+e^ x ) (2+e^ x ) - Vidyadip

Question

Integrate the function $\frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)}$

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Answer

Let, $I=\frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)}$
Let $e^x = t \Rightarrow e^x dx = dt$
$\Rightarrow \int \frac{\mathrm{e}^{\mathrm{x}}}{\left(1+\mathrm{e}^{\mathrm{x}}\right)\left(2+\mathrm{e}^{\mathrm{x}}\right)} \mathrm{d} \mathrm{x}=\int \frac{1}{(1+\mathrm{t})(2+\mathrm{t})} \mathrm{dt}$
= $\int\left[\frac{1}{(1+t)}-\frac{1}{(2+t)}\right] d t$
= $\int\left[\frac{1}{(1+t)}\right] d t-\int\left[\frac{1}{(2+t)}\right] d t$
= $\log |(1+t)|-\log |(2+t)|+c$
= $\log \left|\frac{1+\mathfrak{t}}{2+\mathfrak{t}}\right|+{C}$
$\Rightarrow \mathrm{I}=\log \left|\frac{1+\mathrm{e}^{\mathrm{x}}}{2+\mathrm{e}^{\mathrm{x}}}\right|+\mathrm{C}$

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