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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(e^{2x)-1}

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

- arctan (e^{2 x} - 1) + e^{2 x} - 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u} - 1}{2} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{v - 1}{v} d v

= \frac{1}{2} \cdot \int \frac{v - 1}{v} d v

Apply u-substitution

: \int \frac{2 w ^{2}}{w ^{2} + 1} d w

= \frac{1}{2} \cdot \int \frac{2 w ^{2}}{w ^{2} + 1} d w

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{w ^{2}}{w ^{2} + 1} d w

\frac{w ^{2}}{w ^{2} + 1} = - \frac{1}{w ^{2} + 1} + 1

= \frac{1}{2} \cdot 2 \cdot \int - \frac{1}{w ^{2} + 1} + 1 d w

Apply the Sum Rule:

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

= \frac{1}{2} \cdot 2 (- \int \frac{1}{w ^{2} + 1} d w + \int 1 d w)

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

\int 1 d w = w

= \frac{1}{2} \cdot 2 (- arctan (w) + w)

Substitute back

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

Simplify

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

= - arctan (e^{2 x} - 1) + e^{2 x} - 1

Add a constant to the solution

= - arctan (e^{2 x} - 1) + e^{2 x} - 1 + C

\frac{d}{d x} (\frac{cot ( x )}{e ^{x}})

\frac{\partial}{\partial x} (x y^{2} + x y - x^{2})

2 e^{x}

\int x + x ln (x) d x

\int t sin^{2} (t) d t

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