8.6.5 What Can Go Wrong?

In Example 3 we already saw one thing that can go wrong when we use a CAS to evaluate an indefinite integral: The integral

∫ 1 + 4 ⁢ x 3 d x

represents a family of continuous functions, but these are not elementary functions, that is, they are not expressible in terms of the functions we study in this course. Thus, the response from any CAS will include symbols we don't recognize.

But there are also situations in which the answer can be expressed in elementary functions, yet a CAS cannot find that answer unless we give it some help.

Example 4

Evaluate ∫ cos 2 x dx cos 4 x+ cos 2 x+1 .

Every CAS we know about will report the answer to this problem in terms of complex numbers and non-elementary functions — or will fail to report any answer. You can see the first possibility quickly by sending the problem to Wolfram|Alpha with the button at the right. Simplifying the answer doesn't help much, and differentiating the answer does not make a convincing case that the answer really is an antiderivative.

(Don't take our word for it — try it.)

However, this integral does represent a family of functions that can be expressed in terms of elementary functions, and a CAS can find that answer if we first transform the problem to another form — which we ask you to do in Activity 5.

Note 3 – Domain and range of the integrand

Activity 5

1. The integral in Example 4 looks like a candidate for a substitution such as \(u=\cos^2x\). The tricky part is expressing \(dx\) in terms of \(du\). First compute \(du\) in terms of \(dx\), and then make substitutions for \(\sin\,x\) and \(\cos\,x\) in terms of \(u\). (Remember that \(\sin\,x\) and \(\cos\,x\) are related by a Pythagorean formula.) Simplify your integrand as much as possible. You should find that if you combine all the square roots, some cancellations occur. What is your transformed integral?
   
   
2. Modify your W|A formula from Example 4 (repeated at the right) to evaluate your integral from part a. Observe that the CAS answer is now expressed entirely in elementary functions (no complex numbers, elliptic integrals, or unknown functions). Since \(u\) is an elementary function of \(x\), that confirms that the original integral can be expressed in elementary functions. However, the algebraic task of expressing the answer in terms of \(x\) looks daunting enough that we might want to try a different transformation instead.
   
   
3. Try the substitution \(u=\cos\,x\), and again be careful expressing \(dx\) in terms of \(du\). As in part a, combine the square root factors and simplify as much as possible. What is your transformed integral?
   
   
4. Use the Indefinite Integral tool (or W|A) to evaluate the integral in part c, and express the answer in terms of \(x\). This time the algebra should be easy.
5. Confirm that your answer in part d is indeed an antiderivative by differentiating it. The answer won't look exactly right until you "unsimplify" the simplification steps you did in part c to get rid of some powers of the cosine function.

Activity 5 has several points. First, a CAS is sometimes useful only if you give it a little help in terms of transforming a problem to a different but equivalent problem. Second, every change of variable transforms an integral into some other integral, but even when the CAS can evaluate the integral in terms of elementary functions, some other transformation might produce better (simpler) results. Simplification is an art, and a machine can't always "see" what human eyes and brain can see. Third, as we will explore further in Problem 1, a less obvious transformation might produce an integral that is so simple we don't even need a CAS for the evaluation.

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