What is the integral of (3+6sinh(x))/(x+2cosh(x)) ?

The integral of (3+6sinh(x))/(x+2cosh(x)) is 3ln|x+2cosh(x)|+C

integral of (3+6sinh(x))/(x+2cosh(x))

\int \frac{3 + 6 sinh ( x )}{x + 2 cosh ( x )} d x

3 ln ∣ x + 2 cosh (x) ∣ + C

Solution steps

\int \frac{3 + 6 sinh ( x )}{x + 2 cosh ( x )} d x

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 ln ∣ u ∣

Substitute back

u = x + 2 cosh (x)

= 3 ln ∣ x + 2 cosh (x) ∣

Add a constant to the solution

= 3 ln ∣ x + 2 cosh (x) ∣ + C

t a y l or (1 - x^{2})^{(\frac{1}{2})}

d er i v a t i v e y = 3 x - 5

\frac{\partial}{\partial y} (ln (2 x + y))

\frac{d}{d x} ((x^{2} + y^{2}) e^{x^{2} + y^{2}})

t an g e n t f (x) = \frac{6 x}{x ^{2} + 1}, at x = - 1

What is the integral of (3+6sinh(x))/(x+2cosh(x)) ?
The integral of (3+6sinh(x))/(x+2cosh(x)) is 3ln|x+2cosh(x)|+C