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Magazine Chapter 9: Problem 42
(a) Make an appropriate \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$\int \frac{\sqrt{4-9 x^{2}}}{x^{2}} d x$$
Short Answer
Expert verified
Use substitution \( u = 4 - 9x^2 \), solve and confirm with CAS.
Step by step solution
01
Choose the Substitution
The given integral is \( \int \frac{\sqrt{4-9x^2}}{x^2} \, dx \). To simplify the integral, we choose the substitution \( u = 4 - 9x^2 \). This is because the radical \( \sqrt{4 - 9x^2} \) suggests a connection with a trigonometric identity or transformation.
02
Differentiate and Solve for dx
Differentiate the substitution \( u = 4 - 9x^2 \) with respect to \( x \) to find \( du \). We get \( du = -18x \, dx \), which implies \( dx = \frac{du}{-18x} \).
03
Substitute Into the Integral
Convert the integral in terms of \( u \) and \( dx \). This gives us \( \int \frac{\sqrt{u}}{x^2} \cdot \frac{du}{-18x} \). To simplify, recall \( x^2 = \frac{4-u}{9} \), thus rewrite the integral as \( -\frac{1}{18} \int \frac{\sqrt{u} \cdot 9}{4-u} \, du \). Further simplification leads to \(-\frac{1}{2} \int \frac{\sqrt{u}}{4-u} \, du \).
04
Evaluate the Integral
Now, use the Endpaper Integral Table to find \( \int \frac{\sqrt{u}}{4-u} \, du \). This table or resource should have a suitable entry or require further substitution or trigonometric identities to evaluate the integral.
05
Confirm Using CAS
Use a Computer Algebra System (CAS) to directly evaluate the integral \( \int \frac{\sqrt{4-9x^2}}{x^2} \, dx \). Ensure that the CAS provides an equivalent result to the one obtained from step (a) using substitution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Tables
Integral tables are like powerful dictionaries for calculus. They contain a list of common integrals and their solutions. When you come across an integral, like \( \int \frac{\sqrt{u}}{4-u} \, du \), you can check if it's listed in an integral table. If it is, you can often find the solution without doing complex calculations, as was done in Step 4 of our solution.Why Use Integral Tables?
- They save time. Instead of working through the integration manually, you can look up the solution and use it directly.
- They provide insight into complex integrals that you might not solve easily on your own.
- A graded exercise like ours might require using these tables to check if an entry or a similar form already exists.
Trigonometric Identities
Trigonometric identities help transform complex expressions, especially when radicals (square roots) or rational expressions are involved, just like the transformation when simplifying \( \sqrt{4-9x^2} \). They're tools to relate different trigonometric functions or transform non-trigonometric problems into ones that can be managed with trigonometric knowledge.Common Trigonometric Substitutions
- For expressions like \( \sqrt{a^2 - x^2} \), a common substitution is \( x = a \sin(\theta) \).
- Another useful substitution is \( x = a \cos(\theta) \) for \( \sqrt{a^2 + x^2} \).
CAS (Computer Algebra System)
CAS, or Computer Algebra Systems, are software programs designed to perform symbolic mathematics. When manually solving an integral becomes a headache, a CAS can compute the solution rapidly. They work by understanding mathematical symbols and provide exact solutions where possible.
Key Benefits of Using CAS
- CAS can automatically perform complex algebraic manipulations, solving equations or integrals much faster than manual calculations.
- They reduce human errors in computation.
- You can confirm your hand-derived solutions and ensure they are accurate, as seen in Step 5 of our solution.
Integration Techniques
Integration techniques are methods used to solve integrals. Different techniques suit different types of integrals, just like choosing the right tool for a job. Integration by substitution, used in our original problem, is one such technique. It involves changing variables to simplify the integral, leading to an easier-to-integrate form.
Primary Integration Techniques
- Substitution: Useful when the integral has a function and its derivative. You substitute part of the integral that makes it easier to evaluate.
- Integration by Parts: Based on the product rule of differentiation, used when integrals have products of functions.
- Partial Fraction Decomposition: Helpful for rational functions where you break down a complex fraction into simpler ones that can be integrated individually.
