91Ó°ÊÓ

Hyperbolic functions are analogs of the trigonometric functions and are widely used in calculus. They include hyperbolic sine (\( \sinh(x) \)) and hyperbolic cosine (\( \cosh(x) \)), which are used to describe certain types of curves called hyperbolas.
Hyperbolic cosine is defined as: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
and hyperbolic sine is: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).
These functions have properties similar to their circular counterparts, sine and cosine, but are based on exponential functions.
One key property of hyperbolic functions to remember in integration is that the derivative of \( \sinh(x) \) is \( \cosh(x) \), and vice versa.
These functions also satisfy the identity: \( \cosh^2(x) - \sinh^2(x) = 1 \), which mirrors the Pythagorean identity for trigonometric functions.
Recognizing these functions within integrals allows for different techniques to be implemented effectively.

Integration is a core concept in calculus, used to find the area under a curve, among other applications. Different functions require different techniques for integration.
There are various methods, such as substitution, integration by parts, and partial fraction decomposition, available to tackle difficult integrals.In the context of hyperbolic functions, knowing their derivatives and antiderivatives is crucial.

• The integral of \( \cosh(u) \) with respect to \( u \) is \( \sinh(u) + C \).
• The integral of \( \sinh(u) \) with respect to \( u \) is \( \cosh(u) + C \).

Applying the correct formula can simplify many problems.Apart from formula-based approaches, integral transformations through substitution are often necessary.
Complex functions can sometimes be rewritten into simpler forms that are easier to integrate.
Choosing the right technique depends on the form of the function you are dealing with. Knowing when a substitution can simplify a problem, or when a direct integration is possible, will save time and effort.

The substitution method is a widely used technique to simplify integrals. It is especially helpful when the integral contains composite functions.
By substituting variables, it transforms the integral into a basic form that is easier to evaluate.Consider an integral like \( \int f(g(x))g'(x) \, dx \).
The substitution method involves setting \( u = g(x) \), and \( du = g'(x) \, dx \).
This simplifies the integral to \( \int f(u) \, du \), which is often easier to solve.
In the given exercise, the substitution was set up as \( u = \frac{x}{2} - \ln 3 \).
From this, \( \frac{du}{dx} = \frac{1}{2} \) was derived, leading to \( dx = 2 \, du \).
This transformed the integral:
\( \int 6 \cosh\left(\frac{x}{2} - \ln 3\right) dx \) to \( 12 \int \cosh(u) du \).
By evaluating \( 12 \sinh(u) + C \), and substituting back to the original variable, the original integral is solved. The substitution method simplifies integration significantly and is a key technique in overcoming complicated integral challenges.