Question

For the following exercises, find the antiderivatives for the given functions.\(\cosh (x)+\sinh (x)\)

Short answer

The antiderivative is \( \sinh(x) + \cosh(x) + C \).

Step-by-step solution

Identify the Antiderivatives of Hyperbolic Functions

Recall that the antiderivative of \( \cosh(x) \) is \( \sinh(x) \) and the antiderivative of \( \sinh(x) \) is \( \cosh(x) \). These facts are fundamental and derived from the definitions of the hyperbolic functions. Knowing these will help you find the antiderivatives of the terms in the expression.

Consider the Given Function

The given function is \( \cosh(x) + \sinh(x) \). We need to find its antiderivative. Since we can integrate the functions individually, we will deal with each part separately.

Integrate Each Term Individually

Start by integrating each term:- The integral of \( \cosh(x) \) is \( \sinh(x) \).- The integral of \( \sinh(x) \) is \( \cosh(x) \).Therefore, the function \( \cosh(x) + \sinh(x) \) integrates separately to its antiderivative form.

Combine the Antiderivatives and Add Constant of Integration

By combining the antiderivatives from the previous step, we find:\[\int (\cosh(x) + \sinh(x)) \, dx = \sinh(x) + \cosh(x) + C\]where \( C \) represents the constant of integration, accounting for the family of solutions.

Key concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that resemble trigonometric functions but are based on hyperbolas, rather than circles. The most common hyperbolic functions include the hyperbolic sine, denoted as \( \sinh(x) \), and the hyperbolic cosine, denoted as \( \cosh(x) \). These functions are defined in terms of the exponential function:

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

These functions come with their own identities and properties, similar to the sine and cosine functions in trigonometry. For example, unlike trigonometric identities, one key hyperbolic identity is \( \cosh^2(x) - \sinh^2(x) = 1 \).
Hyperbolic functions are useful in many areas of mathematics, including calculus, as they simplify many expressions and have convenient derivatives and antiderivatives.

Integration

Integration is the mathematical process of finding the antiderivative of a function. It is essentially the reverse process of differentiation. When integrating a function, we are determining the functions that could derive to give the original function.
For example, an important property of the integral is that it can be distributed over addition, meaning: \[ \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \].
This is exactly the process we use when dealing with the function \( \cosh(x) + \sinh(x) \). We calculate the integral of each term separately and then combine them.
Thus, understanding the integration of hyperbolic functions is similar to understanding the integration rules of trigonometric functions but with adjustments to match the properties of hyperbolic functions. This makes the solving process intuitive once you grasp the basics of integration.

Constant of Integration

Whenever you find an antiderivative, you are actually finding a family of functions that share the same derivative. This is because when you differentiate a constant, the result is zero, meaning any constant term in the original function would vanish in differentiation. Hence, when finding antiderivatives, you must add a constant of integration, denoted by \( C \).
It ensures that all possible antiderivatives are represented.

• This constant accounts for the vertical shift that would occur due to any constant added to the function.
• For instance, if the antiderivative of a function is \( F(x) \), the generalized solution is \( F(x) + C \).

In practical terms, the constant of integration is crucial as it allows flexibility in functions and ensures that the solution applies to all potential antiderivatives. Without it, the solution would be incomplete as it wouldn’t account for all possible initial conditions or vertical translations of the antiderivative.