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

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

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

- 2 ln ∣ - e^{x} + 1 ∣ - e^{x} + 1 + C

Solution steps

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

Expand

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

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

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

= - ln ∣ 1 - e^{x} ∣ + 1 - e^{x} - ln ∣ 1 - e^{x} ∣

Simplify

= - 2 ln ∣ - e^{x} + 1 ∣ - e^{x} + 1

Add a constant to the solution

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

\int 10 x (5 x^{2} + 5)^{6} d x

\int sec (t) (2 sec (t) + 5 tan (t)) d t

\int \frac{cos ( 2 x ) e ^{5 x}}{2} d x

\int 12 x + 7 d x

\int 2 csc (x) sec (x) d x

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