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Magazine Chapter 6: Problem 45
In Exercises 45-48, evaluate the given definite integral. \(\int_{-1}^{1} \sinh x d x\)
Short Answer
Expert verified
The definite integral \( \int_{-1}^{1} \sinh x \, dx \) evaluates to 0.
Step by step solution
01
Understand the Function
The function given in the integral is the hyperbolic sine function, \( \sinh x \). The hyperbolic sine function is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \).
02
Find the Antiderivative
To integrate \( \sinh x \), we need to find its antiderivative. The derivative of \( \cosh x \) is \( \sinh x \), so the antiderivative of \( \sinh x \) is \( \cosh x \).
03
Use the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \), then \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a). \] Applying this to \( \int_{-1}^{1} \sinh x \, dx \), where \( F(x) = \cosh x \), gives \[ \cosh(1) - \cosh(-1). \]
04
Simplify the Calculation
Since \( \cosh x = \frac{e^x + e^{-x}}{2} \), we find \( \cosh(1) = \frac{e^1 + e^{-1}}{2} \) and \( \cosh(-1) = \frac{e^{-1} + e^1}{2} \). Notice that \( \cosh(1) = \cosh(-1) \), so their difference is 0.
05
State the Result
Since \( \cosh(1) = \cosh(-1) \), the definite integral \( \int_{-1}^{1} \sinh x \, dx \) evaluates to 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They have fascinating properties and are used in various areas of mathematics, including calculus and complex analysis. The most commonly used hyperbolic functions are:
Hyperbolic functions share similar identities to their trigonometric counterparts, such as \( \cosh^2 x - \sinh^2 x = 1 \).
They are very useful due to their smooth and continuous nature, providing important relationships in hyperbolic geometry and calculus.
- Sinh or Hyperbolic Sine: \( \sinh x = \frac{e^x - e^{-x}}{2} \)
- Cosh or Hyperbolic Cosine: \( \cosh x = \frac{e^x + e^{-x}}{2} \)
- Tanh or Hyperbolic Tangent: \( \tanh x = \frac{\sinh x}{\cosh x} \)
Hyperbolic functions share similar identities to their trigonometric counterparts, such as \( \cosh^2 x - \sinh^2 x = 1 \).
They are very useful due to their smooth and continuous nature, providing important relationships in hyperbolic geometry and calculus.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a central tenet in calculus that links the concept of differentiation and integration. It consists of two parts:
- The first part states that if \( f(x) \) is a continuous real-valued function on a closed interval \([a, b]\), and \( F \) is the antiderivative of \( f \), then the integral of \( f \) over \([a, b]\) is given by \( F(b) - F(a) \).
- The second part states that if \( F \) is an antiderivative of \( f \) on an interval \( I \), then \( \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x) \).
Antiderivative
An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that its derivative is \( f(x) \). It is often represented as:\[\frac{d}{dx}F(x) = f(x)\]The process of finding an antiderivative is called indefinite integration. Unlike differentiation, which outputs a unique result, there are infinitely many antiderivatives for a given function due to the constant of integration, \( C \), which is added to account for all possibilities.For example:
- For \( f(x) = \sinh x \), the antiderivative is \( F(x) = \cosh x + C \), because the derivative of \( \cosh x \) is \( \sinh x \).
Sinh Function
The hyperbolic sine function, or \( \sinh x \), is a key example of a hyperbolic function. Defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \), it mimics the sine function in trigonometry within the context of hyperbolas rather than circles. Unique Properties of \( \sinh x \):
- Unlike the trigonometric sine function, \( \sinh x \) is not periodic.
- It is an odd function, meaning \( \sinh(-x) = -\sinh(x) \).
- Its derivative is the hyperbolic cosine function, \( \cosh x \).
