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Chapter 7: Problem 2

Does a computer algebra system give an exact result for an indefinite integral? Explain.

Short Answer

Expert verified
Answer: A computer algebra system (CAS) may or may not give an exact result for an indefinite integral. This depends on whether the integral can be expressed as a closed-form, simple expression or not. CAS programs have limitations and may struggle with complicated or non-elementary functions. In such cases, they may provide numerical approximations or more complex expressions with special functions.

Step by step solution

01

Introduction to Computer Algebra System (CAS)

A computer algebra system is a type of software program designed for symbolic manipulation of mathematical expressions. It is capable of performing a variety of mathematical operations, such as simplification, differentiation, integration, and solving equations. Some popular CAS programs include Mathematica, Maple, and WolframAlpha.
02

Introduction to Indefinite Integrals

An indefinite integral, also known as an antiderivative, is used to represent the family of functions that are differentiated to obtain the given function. In mathematical notation, it is represented as \[\int f(x)dx = F(x) + C\] where \(f(x)\) is the given function, \(F(x)\) is an antiderivative of the function, and \(C\) is an arbitrary constant representing the constant of integration.
03

Exact Results, Symbolic and Numerical Answers

An exact result refers to a solution that is completely precise and accurate, without any approximations. There are mainly two types of answers a CAS can provide: symbolic and numerical. Symbolic answers are given in a mathematical notation, and numerical answers are given as decimal approximations.
04

CAS and Indefinite Integrals

When it comes to finding an indefinite integral, a CAS aims to provide an exact, symbolic answer. However, not all integrals can be expressed in a simple, closed-form expression. In such cases, the CAS might not be able to provide an exact result. Instead, it may offer a numerical approximation or a more complicated expression involving special functions.
05

Conclusion

In conclusion, a computer algebra system may or may not give an exact result for an indefinite integral. This depends on whether the integral can be expressed as a closed-form, simple expression or not. CAS programs have limitations and may struggle with complicated or non-elementary functions. In such cases, they may provide numerical approximations or more complex expressions with special functions.

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Most popular questions from this chapter

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\pi / 2} \sec \theta d \theta$$

By reduction formula 4 in Section 3 $$\int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C$$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(9-x^{2}\right)^{-2},\left[0, \frac{3}{2}\right]$$

Recall that the substitution \(x=a \sec \theta\) implies that \(x \geq a\) (in which case \(0 \leq \theta<\pi / 2\) and \(\tan \theta \geq 0\) ) or \(x \leq-a\) (in which case \(\pi / 2<\theta \leq \pi\) and \(\tan \theta \leq 0\) ). Evaluate for \(\int \frac{\sqrt{x^{2}-1}}{x^{3}} d x,\) for \(x>1\) and for \(x<-1\)

Suppose that a function \(f\) has derivatives of all orders near \(x=0 .\) By the Fundamental Theorem of Calculus, \(f(x)-f(0)=\int_{0}^{x} f^{\prime}(t) d t\) a. Evaluate the integral using integration by parts to show that $$f(x)=f(0)+x f^{\prime}(0)+\int_{0}^{x} f^{\prime \prime}(t)(x-t) d t.$$ b. Show (by observing a pattern or using induction) that integrating by parts \(n\) times gives $$\begin{aligned} f(x)=& f(0)+x f^{\prime}(0)+\frac{1}{2 !} x^{2} f^{\prime \prime}(0)+\cdots+\frac{1}{n !} x^{n} f^{(n)}(0) \\ &+\frac{1}{n !} \int_{0}^{x} f^{(n+1)}(t)(x-t)^{n} d t+\cdots \end{aligned}$$ This expression is called the Taylor series for \(f\) at \(x=0\).

Evaluate \(\int \frac{d y}{y(\sqrt{a}-\sqrt{y})},\) for \(a > 0\). (Hint: Use the substitution \(u=\sqrt{y}\) followed by partial fractions.)
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