What is the integral of (e^x)/(sqrt(2-e^{-2x))} ?

The integral of (e^x)/(sqrt(2-e^{-2x))} is 1/2 sqrt(2e^{2x)-1}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (e^x)/(sqrt(2-e^{-2x))}

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

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

Solution steps

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

Apply u-substitution

= \int \frac{u}{2 u ^{2} - 1} d u

Apply u-substitution

= \int \frac{1}{4 v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{1}{v} d v

Apply the Power Rule

= \frac{1}{4} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{4} \cdot 2 (2 (e^{x})^{2} - 1)^{\frac{1}{2}}

Simplify

\frac{1}{4} \cdot 2 (2 (e^{x})^{2} - 1)^{\frac{1}{2}} : \frac{1}{2} 2 e^{2 x} - 1

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

Add a constant to the solution

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

What is the integral of (e^x)/(sqrt(2-e^{-2x))} ?
The integral of (e^x)/(sqrt(2-e^{-2x))} is 1/2 sqrt(2e^{2x)-1}+C