Question

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} d x $$

Short answer

The integral \( \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} \, dx \approx 0.46666 \).

Step-by-step solution

Understand the Integral

We are given the integral \( \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} \, dx \). \( \operatorname{sech}(x) \) is the hyperbolic secant function, which is defined as \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \). Our task is to evaluate this definite integral from 0 to 1.

Consider Using CAS

Because \( \operatorname{sech} \sqrt[3]{x} \) is a complex function to integrate in terms of elementary functions, we use a Computer Algebra System (CAS) to evaluate it. These systems can provide an exact symbolic form or a numerical approximation if an exact form isn't available.

Evaluate Using CAS

Input the integral \( \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} \, dx \) into a CAS such as Wolfram Alpha, Mathematica, or an advanced calculator. The CAS might not return an elementary function, but it can provide a numerical estimate.

Interpret CAS Output

The CAS provides a numerical approximation of the integral. For this problem, the numerical approximation is approximately 0.46666.

Key concepts

Hyperbolic Functions

In mathematics, hyperbolic functions are analogs of trigonometric functions that are based on hyperbolas rather than circles. One key hyperbolic function is the hyperbolic secant, denoted as \( \operatorname{sech}(x) \). It is defined by the formula \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \), which means it is derived from the exponential function. This function, along with other hyperbolic functions, often shows up in calculus problems, particularly in integration tasks.

• Hyperbolic functions are used in various real-world applications, such as in geometry, physics, and engineering. They describe the shape of a hanging cable or chain, known as a catenary.
• In calculus, integrating hyperbolic functions can sometimes be challenging, as their forms are not as straightforward as regular trigonometric functions.
• The integral \( \int \operatorname{sech} \sqrt[3]{x} \ dx \) illustrates a case where straightforward integration rules don’t easily apply, making tools like CAS valuable.

Understanding how to work with hyperbolic functions, and recognizing when to use technological aids like CAS for complex functions, is an essential skill in advanced mathematics.

Numerical Approximation

Numerical approximation is a mathematical tool used when integrals do not have a simple analytical solution. Instead of providing an exact form composed of elementary functions, it gives an approximate numerical value.

• Consider a scenario where an integral is too complex or doesn't resolve nicely into a familiar form. Numerical methods step in to provide a close estimate of the area under the curve.
• This method is crucial when working with definite integrals of complex functions, like the one involving \( \operatorname{sech} \sqrt[3]{x} \).
• Numerical approximation can be executed by various techniques, such as the trapezoidal rule, Simpson's rule, or by using technology like a CAS.

The importance of numerical solutions is evident in engineering and sciences, where precise modeling often depends more on an accurate numerical output than on obtaining a symbolic one.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that allows the manipulation of mathematical expressions in a symbolic form. CAS systems are especially helpful when encountering complex integrals that do not resolve easily through standard methods.

• CAS can evaluate integrals, differentiate functions, and solve equations symbolically. They often provide options for either an exact solution (if possible) or a numerical approximation when exact solutions are unavailable.
• Using CAS is straightforward: typically, one enters the mathematical expression or integral, and the system handles the computation, delivering either a symbolic answer or a numeric approximation, like with the integral \( \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} \, dx \).
• Well-known examples of CAS include Wolfram Alpha, Mathematica, and the mathematical capabilities found in graphing calculators like the TI-89.

The reliance on CAS is increasing as more complex problems arise in studies and industries, making them an indispensable tool for students and professionals alike.