50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0

Verified step by step guidance

1. Identify the integral to solve: \(\int x^{n} \cos(ax) \, dx\) where \(a \neq 0\).
2. Choose functions for integration by parts: let \(u = x^{n}\) (which simplifies when differentiated) and \(dv = \cos(ax) \, dx\) (which integrates easily).
3. Compute the derivatives and integrals needed: \(du = n x^{n-1} \, dx\) and \(v = \int \cos(ax) \, dx = \frac{\sin(ax)}{a}\).
4. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so substitute to get \(\int x^{n} \cos(ax) \, dx = x^{n} \cdot \frac{\sin(ax)}{a} - \int \frac{\sin(ax)}{a} \cdot n x^{n-1} \, dx\).
5. Factor constants out of the integral to write the reduction formula: \(\int x^{n} \cos(ax) \, dx = \frac{x^{n} \sin(ax)}{a} - \frac{n}{a} \int x^{n-1} \sin(ax) \, dx\).

Key Concepts

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, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.

Reduction Formulas

Reduction formulas express an integral involving a power or parameter in terms of a similar integral with a lower power or simpler form. They help solve complex integrals step-by-step by reducing the problem to easier cases, often used in integrals involving powers of x and trigonometric functions.

Integration of Trigonometric Functions with Polynomial Factors

Integrals combining polynomials like xⁿ and trigonometric functions such as cos(ax) or sin(ax) require careful application of integration techniques. Recognizing how to handle the polynomial part and the oscillatory trigonometric part is essential, often involving repeated integration by parts to reduce powers.