What is the integral of (5e^s)/(1+3e^s) ?

The integral of (5e^s)/(1+3e^s) is 5/3 ln|1+3e^s|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (5e^s)/(1+3e^s)

\int \frac{5 e ^{s}}{1 + 3 e ^{s}} d s

\frac{5}{3} ln ∣ 1 + 3 e^{s} ∣ + C

Solution steps

\int \frac{5 e ^{s}}{1 + 3 e ^{s}} d s

Take the constant out:

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

= 5 \cdot \int \frac{e ^{s}}{1 + 3 e ^{s}} d s

Apply u-substitution

: \int \frac{1}{3 u} d u

= 5 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 1 + 3 e^{s}

= 5 \cdot \frac{1}{3} ln ∣ 1 + 3 e^{s} ∣

Simplify

5 \cdot \frac{1}{3} ln ∣ 1 + 3 e^{s} ∣ : \frac{5}{3} ln ∣ 1 + 3 e^{s} ∣

= \frac{5}{3} ln ∣ 1 + 3 e^{s} ∣

Add a constant to the solution

= \frac{5}{3} ln ∣ 1 + 3 e^{s} ∣ + C

\int x e^{\frac{2}{5} x^{2} + 7} d x

\int e^{x y} (y cos (x) - sin (x)) d x

\int \frac{arctan ( 2 x )}{x ^{2}} d x

\int (9 - 2 x)^{11} d x

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

What is the integral of (5e^s)/(1+3e^s) ?
The integral of (5e^s)/(1+3e^s) is 5/3 ln|1+3e^s|+C