Integrate the following functions w.r.t. x: e^ 3 z e^ 3 x +1 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$\frac{e^{3 z}}{e^{3 x}+1}$

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Answer

Let $I=\int \frac{e^{3 x}}{e^{3 x}+1} d x$
$\text { Put } e^{3 x}+1=t$
$\therefore 3 e^{3 x} d x=d t$
$\therefore e^{3 x} d x=\frac{d t}{3}$
$\therefore I=\int \frac{1}{t} \cdot \frac{d t}{3}=\frac{1}{3} \int \frac{1}{t} d t$
$=\frac{1}{3} \log |t|+c=\frac{1}{3} \log \left|e^{3 x}+1\right|+c .$

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