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.

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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: \[\int \sin(3x)\sin(5x) \, dx = \frac{1}{2} \int \cos(2x) - \cos(8x) \, dx\]

Key Concepts

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.