Integration of Hyperbolic Functions


	Math24.net Calculus

The 6 basic hyperbolic functions are defined by

There are the following differentiation and integration formulas for hyperbolic functions:

We provide here a list of useful hyperbolic identities:

When an integrand contains a hyperbolic function, the integral can be reduced to integrating a rational function by using the substitution

Example 1

Calculate the integral

Solution.

We make the substitution: u = 2 + 3sinh x, du = 3cosh xdx. Then

. Hence, the integral is

Example 2

Evaluate

Solution.

Since

, and, hence,

, we can write the integral as

Making the substitution u = cosh x, du = sinh xdx, we obtain

Example 3

Evaluate the integral

Solution.

We use integration by parts:

. Let

. Then,

. Hence, the integral is

Example 4

Evaluate the integral

Solution.

Since

, we obtain

Example 5

Find the integral

Solution.

By definition of the hyperbolic cosine,

. Hence, the integral is equal

Example 6

Find the integral

Solution.

By definition,

and

. Hence,

We make the substitution: u = e^x, du = e^xdx and calculate the initial integral:

Example 7

Calculate the integral

Solution.

Applying the formulas

and

, we get

Example 8

Evaluate the integral

Solution.

We use integration by parts. Let

Then

Apply integration by parts again to the latter integral. Let

Then we have

Solving this equation for

, we obtain the complete answer: