What is the integral of (3e^x)/(2-e^x) ?

The integral of (3e^x)/(2-e^x) is -3ln|2-e^x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3e^x)/(2-e^x)

\int \frac{3 e ^{x}}{2 - e ^{x}} d x

- 3 ln ∣ 2 - e^{x} ∣ + C

Solution steps

\int \frac{3 e ^{x}}{2 - e ^{x}} d x

Take the constant out:

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

= 3 \cdot \int \frac{e ^{x}}{2 - e ^{x}} d x

Apply u-substitution

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

= 3 \cdot \int - \frac{1}{u} d u

Take the constant out:

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

= 3 (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- ln ∣ u ∣)

Substitute back

u = 2 - e^{x}

= 3 (- ln ∣ 2 - e^{x} ∣)

Simplify

= - 3 ln ∣ 2 - e^{x} ∣

Add a constant to the solution

= - 3 ln ∣ 2 - e^{x} ∣ + C

\int 3 e^{- t} sin (t) d t

\int x^{2} \cdot ln (x^{3}) d x

\int (7 x + e^{2 x}) d x

\int 15 x^{2} (x^{3} - 7)^{4} d x

\int (\frac{1}{y ^{2} + 1}) d y

What is the integral of (3e^x)/(2-e^x) ?
The integral of (3e^x)/(2-e^x) is -3ln|2-e^x|+C