Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.QGYR.4

Capitolo 8, Problema 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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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.

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The Product Rule

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.

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Introduction to Trigonometric Functions

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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Verifying Trig Equations as Identities Example 4

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