What is the integral of (cosh(x))/(cosh^2(x)-1) ?

The integral of (cosh(x))/(cosh^2(x)-1) is -csch(x)+C

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integral of (cosh(x))/(cosh^2(x)-1)

\int \frac{cosh ( x )}{cosh ^{2} ( x ) - 1} d x

- csch (x) + C

Solution steps

\int \frac{cosh ( x )}{cosh ^{2} ( x ) - 1} d x

Rewrite using trig identities

= \int \frac{cosh ( x )}{sinh ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

= \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{u}

Substitute back

u = sinh (x)

= - \frac{1}{sinh ( x )}

Use the Hyperbolic identity:

\frac{1}{sinh ( x )} = csch (x)

= - csch (x)

Add a constant to the solution

= - csch (x) + C

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What is the integral of (cosh(x))/(cosh^2(x)-1) ?
The integral of (cosh(x))/(cosh^2(x)-1) is -csch(x)+C