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Hyperbolic functions are mathematical functions that, similar to trigonometric functions, describe types of curves. Specifically, the hyperbolic sine function, represented as \( \sinh(x) \), is a fundamental hyperbolic function used frequently in calculus. It can be defined using exponential functions as follows:\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]These functions arise naturally when describing shapes like hyperbolas, just as sinusoidal functions describe circles. They are essential in modeling certain natural phenomena and appear in solutions to hyperbolic differential equations. Understanding the definition and application of the hyperbolic sine function is vital for solving integrals that involve these functions.

• \( \sinh(x) \) is an odd function, meaning \( \sinh(-x) = -\sinh(x) \).
• It is related to \( \cosh(x) \), the hyperbolic cosine function, which is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

These properties make hyperbolic functions useful for various calculations, especially in calculus, where they are integral to solving certain types of integrals.

The substitution method in integral calculus is a technique used to simplify the evaluation of integrals. This involves a change of variables, making a complex integral easier to solve by recognizing a function and its derivative within the integral. This approach is similar to the reverse of the chain rule used in differentiation.For the given integral, the substitution method plays a crucial role. By letting \( u = \frac{x}{5} \), we translate the integral into a more manageable form. This transformation simplifies the integral of a function composed of another, linear function. Here's how it works:

• Identify a part of the integral that can be substituted to simplify it. In this case, it is the linear part \( \frac{x}{5} \).
• Differentiate this substitution to find \( dx \) in terms of \( du \). Here, since \( u = \frac{x}{5} \), then \( x = 5u \), and \( dx = 5 \, du \).

By substituting \( u = \frac{x}{5} \) and \( dx = 5 \, du \) into the integral, you get \( \int \sinh(u) \cdot 5 \, du \), simplifying the integration process. Such strategic substitutions are essential in tackling more complex integrals efficiently.

Integrals in calculus come in two primary varieties: definite and indefinite. An **indefinite integral** represents a family of functions and includes a constant of integration, \( C \), since any constant differentiates to zero. It's expressed as \[ \int f(x) \, dx = F(x) + C,\]where \( F(x) \) is the antiderivative of \( f(x) \).A **definite integral**, on the other hand, is computed over a particular interval \([a, b]\) and represents the net area under the curve between two points. It's given by:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a).\]This net area measurement excludes the constant \( C \) because it cancels out during evaluation.For the discussed integral of \( \sinh \frac{x}{5} \, dx \), as it is not limited to bounds, it is an indefinite integral. The result is \( 5 \cosh \left( \frac{x}{5} \right) + C \), indicating it's part of a family of functions that differ only by a constant. Understanding these types of integrals and their differences is crucial when approaching problems in integral calculus, ensuring clarity on what the resulting function or number represents.