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Hyperbolic functions are analogs of trigonometric functions but for a hyperbola. The main hyperbolic functions are

• \( \sinh(x) \) which is known as the hyperbolic sine, and
• \( \cosh(x) \), the hyperbolic cosine.

They are defined using exponential functions:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

These functions are pivotal in calculus, especially when dealing with certain integrals and differential equations related to hyperbolas.

In the exercise, we focus on integrating \( \sinh(3t) \), which is a common task in calculus when you work with hyperbolic functions.Understanding how these functions relate to their integral and derivative counterparts expands your calculus toolkit significantly.Another key point is that hyperbolic functions, like trigonometric ones, form the basis for expressing more complex calculations.

Differentiation is the process of finding the derivative of a function. In calculus, this means calculating the rate at which a function changes at any given point.

When differentiating hyperbolic functions, specific rules apply:

• The derivative of \( \sinh(ax) \) is \( a \cosh(ax) \).
• Similarly, the derivative of \( \cosh(ax) \) is \( a \sinh(ax) \).

In the provided solution, differentiation is used to confirm the correctness of the integration process.

After integrating \( \sinh(3t) \) to get \( \frac{1}{3} \cosh(3t) + C \), differentiating back checks the work by inversely confirming that it returns to the original function \( \sinh(3t) \).

This is an essential step since it ensures that the integral was calculated correctly by going through the reverse operation.

Integral verification involves checking that the integration of a function produces a result that, when differentiated, returns to the original function. This process gives us confidence in our calculus operations and solutions.

Here’s why this step is crucial:- It acts as a self-check mechanism within integrals and derivatives.- Verifying integrals reinforces understanding of both integration and differentiation concepts.- It helps identify mistakes in manual calculations or interpretations early.

In the exercise, the verification step involves differentiating the integrated result \( \frac{1}{3} \cosh(3t) + C \) and confirming it matches the original function \( \sinh(3t) \).
Using the derivative rules for hyperbolic functions, we get back to the starting point, validating our work.