What is the integral of (e^2x-1)/(e^2x+1) ?

The integral of (e^2x-1)/(e^2x+1) is x-2/(e^2)ln|e^2x+1|+2/(e^2)+C

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integral of (e^2x-1)/(e^2x+1)

integral

\frac{e ^{2} x - 1}{e ^{2} x + 1}

x - \frac{2}{e ^{2}} ln e^{2} x + 1 + \frac{2}{e ^{2}} + C

Solution steps

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

\frac{e ^{2} x - 1}{e ^{2} x + 1} = \frac{2 e ^{2} x}{e ^{2} x + 1} - 1

= \int \frac{2 e ^{2} x}{e ^{2} x + 1} - 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 e ^{2} x}{e ^{2} x + 1} d x - \int 1 d x

\int \frac{2 e ^{2} x}{e ^{2} x + 1} d x = 2 x + \frac{2}{e ^{2}} - \frac{2}{e ^{2}} ln e^{2} x + 1

\int 1 d x = x

= 2 x + \frac{2}{e ^{2}} - \frac{2}{e ^{2}} ln e^{2} x + 1 - x

Simplify

= x - \frac{2}{e ^{2}} ln e^{2} x + 1 + \frac{2}{e ^{2}}

Add a constant to the solution

= x - \frac{2}{e ^{2}} ln e^{2} x + 1 + \frac{2}{e ^{2}} + C

3 x + 1

x (cos (x))^{2}

x^{8} + 9

(9 x + 36)^{- 1}

\frac{y ^{3}}{x ^{4}}

What is the integral of (e^2x-1)/(e^2x+1) ?
The integral of (e^2x-1)/(e^2x+1) is x-2/(e^2)ln|e^2x+1|+2/(e^2)+C