Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(2x) cos(3x) dx

Verified step by step guidance

1. Recognize that the integral involves the product of sine and cosine functions with different arguments: \(\int \sin(2x) \cos(3x) \, dx\).
2. Use the product-to-sum trigonometric identity to rewrite the product of sine and cosine as a sum of sines: \(\sin(A) \cos(B) = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
3. Apply the identity with \(A = 2x\) and \(B = 3x\) to get: \(\sin(2x) \cos(3x) = \frac{1}{2} [\sin(5x) + \sin(-x)]\).
4. Simplify \(\sin(-x)\) using the odd property of sine: \(\sin(-x) = -\sin(x)\), so the expression becomes \(\frac{1}{2} [\sin(5x) - \sin(x)]\).
5. Rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \frac{1}{2} \int [\sin(5x) - \sin(x)] \, dx\), then integrate each sine term separately using the integral formula \(\int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C\).

Key Concepts

Product-to-Sum Trigonometric Identities

These identities transform products of sine and cosine functions into sums or differences of trigonometric functions, simplifying integration. For example, sin(A)cos(B) = ½[sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a form easier to integrate.

Basic Integration of Trigonometric Functions

Integrating sine and cosine functions involves applying standard integral formulas, such as ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these forms allows direct evaluation after simplification.

Use of Integral Tables

Integral tables provide formulas for common integrals, including trigonometric integrals, which can save time and reduce errors. Referring to these tables helps identify the correct integral form and constants, especially for more complex expressions.