Problem 71 Writing Describe the various app... [FREE SOLUTION]

Chapter 7: Problem 71

Writing Describe the various approaches for evaluating integrals of the form $$\int \sin ^{m} x \cos ^{n} x d x$$ Into what cases do these types of integrals fall? What procedures and identities are used in each case?

Short Answer

Expert verified

Use different techniques for each case: odd $ m $, odd $ n $, or both even.

Step by step solution

Understanding Different Cases

Integrals of the form $ \int \sin^{m} x \cos^{n} x \; dx $ are solved based on the parity (odd or even) of $ m $ and $ n $. There are three main cases: 1) When $ m $ is odd; 2) When $ n $ is odd; 3) When both $ m $ and $ n $ are even.

Case 1 - When m is Odd

If $ m $ is odd, we can factor out one $ \sin x $ from the integrand. The remaining $ \sin^{m-1} x $ can then be expressed as $ (1 - \cos^2 x)^{\frac{m-1}{2}} $, allowing for a substitution $ u = \cos x $, $ du = -\sin x \, dx $. This simplifies the integral into a polynomial that can be easily integrated.

Case 2 - When n is Odd

If $ n $ is odd, we factor out one $ \cos x $ from the integrand. Then, the remaining $ \cos^{n-1} x $ can be rewritten as $ (1 - \sin^2 x)^{\frac{n-1}{2}} $. Using the substitution $ u = \sin x $, $ du = \cos x \, dx $, the integral becomes a manageable polynomial.

Case 3 - When Both m and n are Even

When both $ m $ and $ n $ are even, neither $ \sin x $ nor $ \cos x $ can be directly factored out. We use power-reduction identities: $ \sin^2 x = \frac{1 - \cos 2x}{2} $ and $ \cos^2 x = \frac{1 + \cos 2x}{2} $ to lower the powers of $ \sin x $ and $ \cos x $. This often requires multiple applications of these identities to fully simplify and integrate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques

Integration techniques are essential tools for solving trigonometric integrals. They provide a systematic way to find the integral of functions like $ \int \sin^{m} x \cos^{n} x \, dx $. This expression can be tricky, but recognizing it's composed of sine and cosine functions gives us a clear path on how to tackle it, based on whether the powers $ m $ and $ n $ are odd or even.

Substitution Method: One of the most fruitful techniques involves substitution. When either sine or cosine has an odd power, substitution usually leads to simplification. For instance, if $ m $ is odd, substituting $ u = \cos x $ makes the integral easier to handle.
Power-Reduction: When both $ m $ and $ n $ are even, simple substitutions aren't enough. Instead, we use power-reduction identities to transform higher power sine or cosine into multiple lower powers, making the integral approachable.

The trick is always to simplify the expression into a polynomial or a more straightforward trigonometric function, allowing for easy integration.

Odd and Even Functions

When dealing with $ \int \sin^{m} x \cos^{n} x \, dx $, identifying the parity of the exponents $ m $ and $ n $ is crucial. Odd and even functions influence the integration strategy, and knowing which is which determines our approach.

Odd Functions: Consider $ m $ or $ n $ being odd. If $ m $, for example, is odd, $ \sin x $ or $ \cos x $ can be factored out to simplify the integral. This step allows applying trigonometric identities more effectively as it maintains symmetry in substitution.
Even Functions: If $ m $ and $ n $ are even, direct substitution doesn鈥檛 suffice. Instead, power-reduction formulas are needed to simplify the trigonometric functions to lower degrees. This enables a clearer path to integration by slowly converting complex functions into sums of easier to digest components.

Understanding the nature of odd and even functions is like having a map during a complex journey. It guides you towards the right integration path, ensuring that you apply the most efficient techniques and identities.

Power-Reduction Identities

Power-reduction identities are indispensable when both $ m $ and $ n $ are even in

\int \sin^{m} x \cos^{n} x \, dx

. These identities help reduce the exponents of trigonometric functions, making the integral easier to solve.

Sine Power-Reduction: The identity $ \sin^2 x = \frac{1 - \cos 2x}{2} $ transforms $ \sin^2 x $ into something more manageable. It expresses the square of sine in terms of cosine, effectively lowering the power by decomposing it into two simpler parts.
Cosine Power-Reduction: Similarly, the identity $ \cos^2 x = \frac{1 + \cos 2x}{2} $ simplifies $ \cos^2 x $ by expressing it in terms of $ \cos 2x $. This change reduces its power, thereby simplifying the integration process.

Power-reduction identities are powerful because they transform complicated expressions into simpler ones that can be handled more easily. Using these identities repeatedly can break down integrals with high power trigonometric functions into a series of smaller, more approachable steps.

91影视

Writing Describe the various approaches for evaluating integrals of the form $$\int \sin ^{m} x \cos ^{n} x d x$$ Into what cases do these types of integrals fall? What procedures and identities are used in each case?

Short Answer

Step by step solution

Understanding Different Cases

Case 1 - When m is Odd

Case 2 - When n is Odd

Case 3 - When Both m and n are Even

Key Concepts

Integration Techniques

Odd and Even Functions

Power-Reduction Identities

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