What is the integral of (e^xsqrt(e^x-1))/(e^x+3) ?

The integral of (e^xsqrt(e^x-1))/(e^x+3) is 4(-arctan(1/2 sqrt(e^x-1))+1/2 sqrt(e^x-1))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (e^xsqrt(e^x-1))/(e^x+3)

\int \frac{e ^{x} e ^{x} - 1}{e ^{x} + 3} d x

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

Solution steps

\int \frac{e ^{x} e ^{x} - 1}{e ^{x} + 3} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{2} + 4} d u

= \int \frac{2 u ^{2}}{u ^{2} + 4} d u

Take the constant out:

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

= 2 \cdot \int \frac{u ^{2}}{u ^{2} + 4} d u

\frac{u ^{2}}{u ^{2} + 4} = - \frac{4}{u ^{2} + 4} + 1

= 2 \cdot \int - \frac{4}{u ^{2} + 4} + 1 d u

Apply Integral Substitution

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

Take the constant out:

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

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

Apply the Sum Rule:

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

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

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

\int 1 d v = v

= 2 \cdot 2 (- arctan (v) + v)

Substitute back

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

Simplify

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

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

Add a constant to the solution

= 4 (- arctan (\frac{1}{2} e^{x} - 1) + \frac{1}{2} e^{x} - 1) + C

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

n = 1 \sum \infty (\frac{1}{8})^{n}

\frac{d}{d x} (\frac{x + 3}{x ^{3} + x - 2})

\int \frac{1}{( 8 x - 1 ) ^{3}} d x

\int \frac{2 x ^{2} - 1}{( 4 x - 1 ) ( x ^{2} + 1 )} d x

What is the integral of (e^xsqrt(e^x-1))/(e^x+3) ?
The integral of (e^xsqrt(e^x-1))/(e^x+3) is 4(-arctan(1/2 sqrt(e^x-1))+1/2 sqrt(e^x-1))+C