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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (e^{2x}+1)/(e^x+1)

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

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

Solution steps

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

Expand

\frac{e ^{2 x} + 1}{e ^{x} + 1} : \frac{e ^{2 x}}{e ^{x} + 1} + \frac{1}{e ^{x} + 1}

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

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

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

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

Simplify

= e^{x} + 1 - 2 ln ∣ e^{x} + 1 ∣ + x

Add a constant to the solution

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

\int sin (2 x) sin (3 x) cos (4 x) d x

\int \frac{1}{x ^{2} + 3 x - 10} d x

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

\int \frac{1}{x ^{2} ( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

\int x^{2} cos (5 x^{3} - 1) d x

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