Question

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

Answer 1

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

Dividing Numerator and Denominator by $e^x$

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

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

Taking Denominator $e^x+\frac{1}{e^x}=t$

Diferrentiating both sides $(e^x-\frac{1}{e^x})dx=dt$

Now intergration becomes, $\int \frac{dt}{t}$

$\to ln\left(t\right)+C$

Putting $t$ we get,$\int \frac{e^{2x}-1}{e^{2x}+1}dx=ln(e^x+\frac{1}{e^x})+C$