Advanced Integration Techniques

Aug 29, 2025

Overview

This lecture explores advanced integration techniques, focusing on integration by parts and the derivation and application of reduction formulas for powers of sine functions. The session emphasizes when to use each method, how to derive the formulas, and strategies for solving more complex integrals, including those involving products and higher powers of trigonometric functions.

Integration Review & Motivation

  • When approaching an integral:
    • First, check if it matches a form in your integration table for a straightforward solution.
    • If not, attempt substitution (u-substitution), which is essentially the inverse of the chain rule.
    • If substitution does not work, move on to more advanced techniques such as integration by parts.
  • Integration by parts is particularly useful for products of functions where substitution fails.
  • The lecture also introduces reduction formulas, which are especially helpful for integrals involving higher powers of trigonometric functions.
  • Always follow a logical order: check the table, try substitution, then use integration by parts, and finally consider other advanced methods if needed.

Derivation of Integration by Parts

  • Integration by parts is based on the product rule for derivatives:
    • Product rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
    • Integrating both sides: ∫f'(x)g(x) dx + ∫f(x)g'(x) dx = f(x)g(x)
    • Rearranged: ∫f(x)g'(x) dx = f(x)g(x) − ∫f'(x)g(x) dx
  • Using substitution:
    • Let u = f(x), v = g(x), du = f'(x) dx, dv = g'(x) dx
    • The formula becomes: ∫u dv = u·v − ∫v du
  • This formula allows you to transform a difficult integral into a potentially easier one by shifting the complexity from one part of the integrand to another.
  • The derivation shows that integration by parts is essentially the "inverse" of the product rule for derivatives.

Strategies for Choosing u and dv

  • Choose u so that its derivative (du) simplifies the integral, ideally reducing the degree or complexity (e.g., polynomials, ln x).
  • Choose dv so that it is easy to integrate to get v.
  • Good choices for u often include:
    • Logarithmic functions (e.g., ln x)
    • Algebraic functions (e.g., x^n)
    • Inverse trigonometric functions
  • dv should be the remaining part of the integrand that is easily integrable.
  • If the resulting integral ∫v du is more complicated than the original, switch your choices for u and dv.
  • The goal is to reduce the complexity of the integral with each application.
  • If the integral does not get simpler or the new integral is harder, reconsider your choices for u and dv.

Integration by Parts – Examples

  • Example 1: ∫x eˣ dx
    • Let u = x (derivative simplifies), dv = eˣ dx (easy to integrate)
    • Result: x eˣ − eˣ + C
  • Example 2: ∫ln x dx
    • Let u = ln x, dv = dx
    • Result: x ln x − x + C
  • Example 3: ∫x³ sin 2x dx
    • Use integration by parts repeatedly, each time reducing the power of x by one.
    • Continue until the remaining integral is basic and can be solved directly.
  • Example 4: ∫x³ ln x dx
    • Choose u = ln x, dv = x³ dx
    • Apply integration by parts, then use substitution or further integration by parts as needed.
  • Example 5: ∫x³ cos(2x) dx
    • Apply integration by parts multiple times, reducing the power of x with each step.
    • The process may require several iterations before reaching a basic integral.

Integrals Involving Products with Trigonometric/Exponential Functions

  • For integrals like ∫eˣ sin 2x dx or ∫eˣ cos 2x dx:
    • Apply integration by parts twice.
    • The original integral may reappear in the process.
    • Set up an equation with the original integral on both sides and solve algebraically.
    • Example: ∫eˣ sin 2x dx = (eˣ(sin 2x − 2 cos 2x))/5 + C
  • This technique is useful when the process cycles back to the original integral.
  • If the same integral appears on both sides of the equation, add or subtract to isolate the integral and solve for it.

Integrals Involving Powers of Trig Functions (Reduction Formula)

  • The reduction formula expresses an integral with a high power of sine in terms of a lower power.
  • For n ≥ 2:
    • ∫sinⁿx dx = –(1/n) sinⁿ⁻¹x cos x + ((n–1)/n) ∫sinⁿ⁻²x dx + C
  • To use the reduction formula:
    • Apply it repeatedly, each time reducing the exponent by 2.
    • Continue until you reach an integral you can solve directly (e.g., ∫sin x dx).
  • The reduction formula is derived using integration by parts and trigonometric identities.
  • This method is especially helpful for integrals like ∫sin⁴x dx or ∫sin⁶x dx.
  • The same approach can be adapted for other trigonometric powers, such as cosine, by deriving a similar reduction formula.

Advanced Example: Integrals with Recursion and Multiple Parts

  • For more complex integrals, such as ∫sec⁵x dx:
    • Break the integrand into a product (e.g., sec³x · sec²x).
    • Use integration by parts, choosing u and dv so that the resulting integral is manageable.
    • You may need to use trigonometric identities (e.g., tan²x = sec²x − 1) to simplify the integrand.
    • The process may involve recursion, where the original integral reappears and can be solved algebraically by adding or subtracting terms.
    • This approach can be extended to other powers and combinations of trigonometric functions.

Key Terms & Definitions

  • Integration by Parts: A technique for integrating products of functions, based on the formula ∫u dv = u·v − ∫v du.
  • Reduction Formula: A recursive formula that reduces the power of a function in an integral, allowing you to express ∫sinⁿx dx in terms of ∫sinⁿ⁻²x dx.
  • u-substitution: A method where you let u = f(x) to simplify the integral, often used when the derivative of u appears elsewhere in the integrand.
  • Recursion in Integration: When an integral reappears during the process, allowing you to solve for it algebraically.
  • Trigonometric Identities: Equations like tan²x = sec²x − 1, used to simplify integrals involving trigonometric functions.

Action Items / Next Steps

  • Practice integration by parts with a variety of functions, including polynomials, exponentials, logarithms, and trigonometric functions.
  • Work through problems using the reduction formula for powers of sine, and try to derive a similar formula for cosine.
  • Memorize the integration by parts formula and the reduction formula for sine.
  • Review your integration table and practice identifying which technique to use for different types of integrals.
  • Read Section 7.2 for additional integration methods, including trigonometric substitution and more advanced techniques.
  • Try challenging integrals that require multiple applications of integration by parts or recursion, and pay attention to when the original integral reappears.
  • Focus on recognizing patterns in integrals that suggest the use of reduction formulas or algebraic manipulation to solve for the original integral.

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