What is the integral of (cosh(x))/(2+3sinh(x)) ?

The integral of (cosh(x))/(2+3sinh(x)) is 1/3 ln|2+3sinh(x)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (cosh(x))/(2+3sinh(x))

\int \frac{cosh ( x )}{2 + 3 sinh ( x )} d x

\frac{1}{3} ln ∣ 2 + 3 sinh (x) ∣ + C

Solution steps

\int \frac{cosh ( x )}{2 + 3 sinh ( x )} d x

Apply u-substitution

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 2 + 3 sinh (x)

= \frac{1}{3} ln ∣ 2 + 3 sinh (x) ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ 2 + 3 sinh (x) ∣ + C

What is the integral of (cosh(x))/(2+3sinh(x)) ?
The integral of (cosh(x))/(2+3sinh(x)) is 1/3 ln|2+3sinh(x)|+C