Integrating expressions involving hyperbolic functions

Integrating Hyperbolic Functions

Basics of Hyperbolic Functions

Hyperbolic functions, denoted as sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x), are closely related to the exponential functions e^x and e^-x.
They have numerous properties similar to those of the standard trigonometric functions but they are not periodic and they grow exponentially.

Integrating Hyperbolic Functions

Integration is the reverse process of differentiation. To integrate hyperbolic functions, we need to know their derivatives.
The derivative of sinh(x) is cosh(x) and the derivative of cosh(x) is sinh(x).
The derivative of tanh(x) is sech²(x), the derivative of coth(x) is -csch²(x).
Likewise, the derivative of sech(x) is -sech(x) tanh(x) and the derivative of csch(x) is -csch(x) coth(x).

Integrals of Hyperbolic Functions

The integral of sinh(x) is cosh(x) + C, where C is the constant of integration.
The integral of cosh(x) is sinh(x) + C.
The integral of tanh(x) is **ln sech(x) + C** or **ln cosh(x) + C**.
The integral of coth(x) is **ln sinh(x) + C**.
The integral of sech(x) is arctanh(x) + C or arcsinh(x) + C, depending on the preferences of the person doing the maths.
Similarly, the integral of csch(x) is arccoth(x) + C or arccos(x) + C.

Tips for Successful Integration

Always remember that when differentiating or integrating a constant times a function, the constant can be treated as if it’s not there.
Be sure to remember and recognize the derivatives and integrals of the common hyperbolic functions.
Practice problems which involve integration of applications of hyperbolic functions to gain mastery.

Integration by Substitution

Sometimes, the standard integration techniques might not directly apply, and you might have to use integration by substitution.
The substitution method involves finding a suitable substitution to simplify the integrand.
The same method can be used for integrating hyperbolic functions, especially when the integrand is a product of functions.

Other Techniques

Occasionally, it may be helpful to use other tools like integration by parts, especially when integrating products of hyperbolic functions.
The integration by parts formula is ∫ u dv = uv - ∫ v du.
In general, practice is the key for mastering the integration of hyperbolic functions. Whether using basic techniques or more advanced ones like substitution or integration by parts, familiarity with the process is crucial for success.