Mathematics

$$\displaystyle \int { \cfrac { dx }{ \sqrt { { e }^{ 2x } } -1 }  } $$ equals to


ANSWER

$$\ln \left|e^x-1\right|-x+C$$


SOLUTION
Given,

$$\displaystyle \int \dfrac{1}{\sqrt{e^{2x}}-1}dx$$

$$\displaystyle=\int \dfrac{1}{e^x-1}dx$$

$$\displaystyle=\int \dfrac{e^x}{e^x-1}-1dx$$

as $$\displaystyle\dfrac{1}{e^x-1}=\dfrac{e^x}{e^x-1}-1$$

$$\displaystyle=\int \dfrac{e^x}{e^x-1}dx-\int \:1dx$$  

$$=\ln \left|e^x-1\right|-x+C$$
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Single Correct Medium Published on 17th 09, 2020
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