What is the integral of (sinh(x))/(1-4cosh(x)) ?

The integral of (sinh(x))/(1-4cosh(x)) is -1/4 ln|1-4cosh(x)|+C

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integral of (sinh(x))/(1-4cosh(x))

\int \frac{sinh ( x )}{1 - 4 cosh ( x )} d x

- \frac{1}{4} ln ∣ 1 - 4 cosh (x) ∣ + C

Solution steps

\int \frac{sinh ( x )}{1 - 4 cosh ( x )} d x

Apply u-substitution

: \int - \frac{1}{4 u} d u

= \int - \frac{1}{4 u} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{4} ln ∣ u ∣

Substitute back

u = 1 - 4 cosh (x)

= - \frac{1}{4} ln ∣ 1 - 4 cosh (x) ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ 1 - 4 cosh (x) ∣ + C

x \to 0 lim ((x - 1)^{\frac{1}{2}})

(2 x y^{2} + 3 x^{2} y) d x + (2 x^{2} y + x^{3}) d y = 0

(- 5, 3), (5, - 3)

f (x) = x^{2} + 4

\frac{d}{d x} (16 - 4 sin^{2} (7 π x))

What is the integral of (sinh(x))/(1-4cosh(x)) ?
The integral of (sinh(x))/(1-4cosh(x)) is -1/4 ln|1-4cosh(x)|+C