Integration of Hyperbolic Functions


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^{ x}dx
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:

