Tutorial 3: Trigonometric Integrals

Tutorial problems covering integration of trigonometric functions and their powers.

Sections

Powers of Sine and Cosine (Problems 1-4)

  • Odd powers: save one factor, convert rest
  • Even powers: use half-angle formulas
  • Mixed powers: appropriate substitutions

Powers of Tangent and Secant (Problems 5-8)

  • $\int \tan^n x dx$ using $\tan^2 x = \sec^2 x - 1$
  • $\int \sec^n x dx$ (reduction or special techniques)
  • Products $\int \tan^m x \sec^n x dx$

Product-to-Sum Applications (Problems 9-12)

  • Integrals of $\sin(mx)\cos(nx)$
  • Integrals of $\sin(mx)\sin(nx)$
  • Integrals of $\cos(mx)\cos(nx)$

Key Identities Used

Pythagorean:

  • $\sin^2 x + \cos^2 x = 1$
  • $\tan^2 x + 1 = \sec^2 x$

Half-Angle:

  • $\sin^2 x = \frac{1 - \cos 2x}{2}$
  • $\cos^2 x = \frac{1 + \cos 2x}{2}$

Product-to-Sum:

  • $\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]$

Links