$\int \sqrt{e^{2 x} - 1} d x$ is equal to

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Solution

$G i v e n, I = \int \sqrt{e^{2 x} - 1} d x L e t, t = \sqrt{e^{2 x} - 1} \Rightarrow t^{2} = e^{2 x} - 1 \Rightarrow e^{2 x} = t^{2} + 1 D i f f e r e n t i a t i n g b o t h s i d e s \Rightarrow 2 t d t = 2 e^{2 x} d x \Rightarrow d x = \frac{t}{t^{2} + 1} d t S u b s t i t u t i n g t h e v a l u e s i n t h e i n t e g r a l I = \int \frac{t^{2}}{t^{2} + 1} d t \Rightarrow I = \int \frac{t^{2} + 1 - 1}{t^{2} + 1} d t \Rightarrow I = \int 1 - \frac{1}{t^{2} + 1} d t \Rightarrow I = t - {tan}^{- 1} t + C \Rightarrow I = \sqrt{e^{2 x} - 1} - {tan}^{- 1} \sqrt{e^{2 x} - 1} + C$

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