54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:
55. ∫ x² cos(5x) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int x^{2} \cos(5x) \, dx\).
2. Recall that integration by parts is a useful technique here, especially since the integrand is a product of a polynomial and a trigonometric function. The formula for integration by parts is: \(\int u \, dv = uv - \int v \, du\).
3. Choose \(u = x^{2}\) (which simplifies upon differentiation) and \(dv = \cos(5x) \, dx\) (which can be integrated easily). Then compute \(du = 2x \, dx\) and find \(v\) by integrating \(dv\): \(v = \int \cos(5x) \, dx\).
4. Apply the integration by parts formula: \(\int x^{2} \cos(5x) \, dx = x^{2} v - \int v (2x) \, dx\). This will reduce the power of \(x\) in the integral.
5. If the resulting integral still involves a product of \(x\) and a trigonometric function, apply integration by parts again using the same strategy until the integral is fully evaluated.

Key Concepts

Reduction Formulas

Reduction formulas are recursive relationships that express an integral involving a power or function in terms of a simpler integral. They simplify complex integrals by reducing the power or complexity step-by-step, making evaluation manageable.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often used to derive reduction formulas or directly solve integrals like ∫ x² cos(5x) dx.

Trigonometric Integrals

Trigonometric integrals involve integrating functions containing sine, cosine, or other trig functions. Understanding their properties and how they interact with polynomial terms is essential for applying reduction formulas and integration techniques effectively.