Trigonometric Integral Techniques

Overview

This lecture explains how to solve integrals that involve multiplying and raising trigonometric functions (like sine, cosine, tangent, and secant) to powers. It covers step-by-step methods, important identities, and what to do when the usual patterns don’t work.

Strategies for Integrals of Powers of Sine and Cosine

• There are two main cases:
  • Case 1: At least one power is odd
    • Take out one factor of the function with the odd power (either sine or cosine).
    • Use the Pythagorean identity to change the rest into the other function.
    • Example: For sin³x cos²x, take out one sine (sin x), rewrite sin²x as (1 - cos²x), then use substitution.
  • Case 2: Both powers are even
    • Use half-angle (power-reduction) formulas to lower the powers.
    • Example: For sin⁴x cos⁴x, rewrite as (sin x cos x)⁴, use the double-angle identity, and then reduce the powers further.

Substitution and Distribution Techniques

• After using identities to rewrite the integral, use substitution (u-substitution) when possible.
• Only distribute (expand) after substitution to keep things simple.
• Always pay attention to the inside of the function (the argument), especially when using the chain rule.

Integrals Involving Tangent, Secant, Cotangent, and Cosecant

• If the power of tangent or cotangent is odd:
  • Take out one tangent/cotangent and one secant/cosecant.
  • Change the rest to secant/cosecant using identities.
  • Substitute to solve.
• If the power of secant or cosecant is even:
  • Take out a secant²/cosecant².
  • Change the rest to tangent/cotangent using identities.
  • Substitute to solve.
• The main idea is to match the leftover part of the integral with the derivative of your substitution.

Non-Standard Trigonometric Integrals

• If the integral doesn’t fit the usual patterns (like if the angles are different or it’s not a product), rewrite everything in terms of sine and cosine.
• Use trig identities to simplify, and look for any identity that matches the integrand.

Integrals with Different Angles (Product-to-Sum Formulas)

• When you have products like sin(mx)sin(nx), sin(mx)cos(nx), or cos(mx)cos(nx), use product-to-sum identities to turn the product into a sum or difference of single trig functions.

Key Terms & Definitions

• Pythagorean Identity: sin²x + cos²x = 1.
• Half-Angle (Power-Reduction) Formulas:
  • sin²x = ½(1 − cos2x)
  • cos²x = ½(1 + cos2x)
• u-Substitution: Replace part of the integral with u = f(x) and change dx to du.
• Product-to-Sum Identities: Change products of sines and cosines with different angles into sums, for example:
  • sinA cosB = ½[sin(A+B) + sin(A−B)]

Action Items / Next Steps

• Practice problems for each case (odd/even powers, product-to-sum).
• Memorize the main trig identities and product-to-sum formulas.
• Finish homework on trigonometric integrals.
• Review the reduction formula for powers of sine/cosine and substitution steps.
• Watch the lecture video again if you need more help.