Integrals of Trigonometric Functions

Overview

This lesson from Calculus 2 covers Trigonometric Integrals, focusing on strategies for integrating functions involving sine, cosine, tangent, and secant. The main emphasis is on handling integrals with even and odd powers of these functions, using key identities and substitution methods.

Basic Review and Key Formulas

• Derivatives of Trigonometric Functions:

  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec²x
  • d/dx (sec x) = sec x · tan x

• Pythagorean Identity:

  • sin²x + cos²x = 1

• Derived Identities:

  • Dividing both sides by cos²x:
    • tan²x + 1 = sec²x
  • Dividing both sides by sin²x:
    • 1 + cot²x = csc²x

• Double Angle Formulas:

  • sin²x = ½ (1 - cos 2x)
  • cos²x = ½ (1 + cos 2x)
  • For sin²(2x) and cos²(2x), double the angle accordingly.

• Other Useful Identities:

  • sec²x = 1 + tan²x
  • tan²x = sec²x - 1

Integrals of sin and cos

• Odd Power Case:

  • If either sin or cos has an odd power, take one factor out and use substitution for the remaining even power.
  • Example: For ∫sin²x · cos³x dx, take one cos x out, let u = sin x, and use the identity cos²x = 1 - sin²x.

• Even Power Case:

  • If both powers are even, use double angle identities to rewrite the integrand in terms of cos 2x or sin 2x.
  • Example: For ∫sin²x · cos²x dx, use sin²x = ½ (1 - cos 2x) and cos²x = ½ (1 + cos 2x).

• General Approach:

  • Always convert all terms to a single trigonometric function using substitution and identities to simplify the integral.

Integrals of tan and sec

• Even Power of sec:

  • If sec has an even power (e.g., sec⁴x), set u = tan x, since d/dx (tan x) = sec²x.
  • Use sec²x = 1 + tan²x to rewrite higher powers.

• Odd Power of sec and Odd Power of tan:

  • If both sec and tan have odd powers, set u = sec x, since d/dx (sec x) = sec x · tan x.
  • Use tan²x = sec²x - 1 to rewrite tan powers in terms of sec.

• Key Identity Used:

  • sec²x = 1 + tan²x

Integration by Parts

• For Integrals like ∫sec³x dx:
  • Use integration by parts:
    • Let u = sec x, dv = sec²x dx
    • du = sec x tan x dx, v = tan x
  • Apply the formula: ∫u dv = u·v - ∫v du
  • Use identities to simplify the resulting integrals.

Systematic Steps for Solving

• For ∫sin^m(x) · cos^n(x) dx:

  • If either m or n is odd, take one factor out and use substitution.
  • If both are even, use double angle identities.

• For ∫tan^m(x) · sec^n(x) dx:

  • If n is even, set u = tan x.
  • If both m and n are odd, set u = sec x.
  • Always use the identity sec²x = 1 + tan²x to simplify.

Key Terms & Definitions

• Pythagorean Identity: sin²x + cos²x = 1
• Double Angle Identity: sin²x = ½ (1 - cos 2x), cos²x = ½ (1 + cos 2x)
• Integration by Parts: ∫u dv = u·v - ∫v du
• u-substitution: Replacing a function with u to simplify integration

Action Items / Next Steps

• Practice at least 10 problems from the recommended exercises.
• Memorize all the listed trigonometric identities and formulas.
• Apply these methods to textbook examples for hands-on practice.