Evaluate the integrals in Exercises 53–58.
∫ 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 identity for sine and cosine: \(\sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
3. Apply the identity with \(A = 2x\) and \(B = 3x\) to rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \int \frac{1}{2} [\sin(2x + 3x) + \sin(2x - 3x)] \, dx\).
4. Simplify the arguments inside the sine functions: \(\sin(5x)\) and \(\sin(-x)\), so the integral becomes \(\frac{1}{2} \int [\sin(5x) + \sin(-x)] \, dx\).
5. Recall that \(\sin(-x) = -\sin x\), then split the integral into two simpler integrals: \(\frac{1}{2} \left( \int \sin(5x) \, dx - \int \sin x \, dx \right)\), and proceed to integrate each term separately.

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) = 1/2 [sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a more manageable form.

Basic Integration of Sine and Cosine Functions

Integrating sine and cosine functions involves reversing differentiation: ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these formulas allows direct integration once the integrand is simplified.

Constant Multiple Rule in Integration

This rule states that constants can be factored out of integrals, simplifying calculations. For example, ∫a·f(x) dx = a ∫f(x) dx. Applying this rule is essential when coefficients appear after using trigonometric identities.