Module 7: Techniques of Integration III

Module 7: Techniques of Integration III

1. ODD & POSITIVE

   • SINE
     • Take out cosine & use identity: sin^2 x = 1- cos^2x
   • COSINE
     • Take out cosine & use identity: cos^2 x = 1- sin^2x

2. BOTH EVEN & POSITIVE

   • SINE & COSINE
     • Use formulas of half an angle: sin^2 x = 1/2 (1 - cos2x) & cos^2 x = 1/2 (1 + cos2x)
       • Covert it into an odd cosine

3. SECANT EVEN & POSITIVE

   • Save a secant-squared factor & convert remaining factors to tangents, expand & integrate.

4. TANGENT ODD & POSITIVE

   • Save secant-tangent factor & convert remaining factors to secants, expand integrate

5. NO SECANTS & TANGENT IS EITHER ODD OR EVEN POSTIVE

   • Convert a tangent-squared factor & to a secant-squared factor, expand & repeat if necessary

6. NO TANGENT & SECANT POWER ODD BIGGER OR EQUAL TO 3

   • Use integration by parts
     • If none of the first four cases applies, try converting to sines & cosines.

7. For integrals involving cosecants & cotangents, follow similar strategies as integrals with secants & tangents..

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