Techniques of Integration III

Techniques of Integration III

1. Sine and Cosine

   • Sine odd and positive
     • 1.Take out sine 2.Use the identity: sin²x=1-cos²x 3.Use substitution to solve the integral
   • Cosine odd and positive
     • 1.Take out cosine 2.Use the identity: sin²x=1-cos²x 3.Use substitution to solve the integral
   • Sine or cosine even and positive
     • 1.Use formulas of half angles: sin²x=(1-cos2x)/2 cos²x=(1+cos2x)/2 2.Convert it to an odd cosine 3.Solve the integral

2. Tangent and Secant

   • Secant even and positive
     • 1.Save a secant-squared factor and convert the remaining factors to tangents 2.Expand and integraate
   • Tangent odd and positive
     • 1.Save a secant-tangent factor and convert the remaining factors to secants 2. Expand and integrate
   • No secant factors and the power of the tangent is even or odd and positive
     • 1.Convert a tangent-squared factor to a secant-squared factor 2.Expand and integrate
   • No tangent factors and the power of the secant is odd and vbigger or equal than 3
     • 1.Use integration by parts
   • If none of these works, try converting to sines and cosines

3. For integrals involving powers of cotangents ans cosecants, you can follow a strategy similar to that used for powers of tangents and secants