What is the integral of coth(x) ?

The integral of coth(x) is ln|sinh(x)|+C

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integral of coth(x)

\int coth (x) d x

ln ∣ sinh (x) ∣ + C

Solution steps

\int coth (x) d x

Rewrite using trig identities

= \int \frac{cosh ( x )}{sinh ( x )} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = sinh (x)

= ln ∣ sinh (x) ∣

Add a constant to the solution

= ln ∣ sinh (x) ∣ + C

\int \frac{t}{t ^{3}} d t

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

\frac{d}{d x} ((2 cos (2 x) + sin (3 x) + 3)^{2})

y = \frac{A x ^{2}}{x ^{2} - 1}