Question

Evaluate: $\int \frac{e^{2x}-1}{e^{2x}+1}$

Answer 1

$\int \frac{e^{2x}-1}{e^{2x}+1}dx$

Multiply by $e^{-x}$

$\int \frac{(e^{2x}-1)e^{-x}}{(e^{2x}+1)e^{-x}}dx=\int \frac{e^x-e^{-x}}{e^x+e^{-x}}dx$

Let $e^x+e^{-x}=t$

$(e^x-e^{-x})dx=dt$

$\int \frac{(e^x-e^{-x})}{t}\times \frac{dt}{(e^x-e^{-x})}=\int \frac{dt}{t}=ln\left|t\right|+c$

$ln|e^x+e^{-x}|+c$