Question

Evaluate: $\int \frac{e^x}{e^x+1}dx$.

Answer 1

$\Rightarrow \int \frac{e^x+1-1}{e^x+1}dx$

$\Rightarrow \int dx-\int \frac{1}{e^x+1}dx$

$\Rightarrow x-I$

Here I $=\int \frac{1}{e^x+1}dx$

Let $e^x+1=t\Rightarrow e^{x}dx=dt$

$\Rightarrow I=\int \frac{1}{t(t-1)}dt$

$\Rightarrow I=\int \frac{t-(t-1)}{t(t-1)}dt$

$\Rightarrow I=\int \frac{dt}{t-1}-\int \frac{dt}{t}$

$\Rightarrow I=ln(t-1)-ln\left(t\right)$

$\Rightarrow I=ln\left(e^x\right)-ln(e^x+1)$

$\Rightarrow I=x-ln(e^x+1)$

$\Rightarrow x-I=x-(x-ln(e^x+1\left)\right)$

$=ln(e^x+1)$