Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.QGYR.4

Chapter 8, Problem 8.QGYR.4

4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.

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To evaluate integrals involving products of sine and cosine functions like \(\sin(mx)\sin(nx)\), \(\sin(mx)\cos(nx)\), and \(\cos(mx)\cos(nx)\), we use trigonometric product-to-sum identities to simplify the integrand into sums of single trigonometric functions. This makes the integral easier to solve.

For \(\sin(mx)\sin(nx)\), use the identity: \[\sin(mx)\sin(nx) = \frac{1}{2} \left[ \cos((m-n)x) - \cos((m+n)x) \right]\] This transforms the product into a difference of cosines.

For \(\sin(mx)\cos(nx)\), use the identity: \[\sin(mx)\cos(nx) = \frac{1}{2} \left[ \sin((m+n)x) + \sin((m-n)x) \right]\] This converts the product into a sum of sines.

For \(\cos(mx)\cos(nx)\), use the identity: \[\cos(mx)\cos(nx) = \frac{1}{2} \left[ \cos((m-n)x) + \cos((m+n)x) \right]\] This changes the product into a sum of cosines.

Example: To integrate \(\int \sin(3x)\sin(5x) \, dx\), substitute using the first identity: \[\sin(3x)\sin(5x) = \frac{1}{2} \left[ \cos(2x) - \cos(8x) \right]\] Then integrate term-by-term: \[\int \sin(3x)\sin(5x) \, dx = \frac{1}{2} \int \cos(2x) - \cos(8x) \, dx\]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product-to-Sum Formulas

Product-to-sum formulas transform products of sine and cosine functions into sums or differences of trigonometric functions. This simplification is essential for integrating expressions like sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx), making the integrals easier to evaluate.

Trigonometric Substitution in Integration

Trigonometric substitution involves replacing products of sine and cosine functions with their equivalent sum or difference forms using identities. This method reduces complex integrals into simpler ones involving single trigonometric functions, which are straightforward to integrate.

Examples of Applying Product-to-Sum Identities

Applying product-to-sum identities to specific integrals, such as ∫sin(mx)sin(nx)dx, ∫sin(mx)cos(nx)dx, and ∫cos(mx)cos(nx)dx, demonstrates the substitution process. For example, sin(mx)sin(nx) = ½[cos(m−n)x − cos(m+n)x], which simplifies the integral into a sum of cosines.

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Related Practice

Textbook Question

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

Textbook Question

You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.

You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?

(Remember that for Simpson’s Rule the number of subintervals must be even.)

Textbook Question

Evaluate the integrals in Exercises 33–36.

∫ [1 / (x(9 - x²))] dx

Textbook Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (2 − cosx + sinx) / sin²x dx

Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (x² + 1) dx

Textbook Question

Evaluate ∫ sec θ dθ by:

a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.