Trigonometric Integrals
Integrating powers of Sine and Cosine requires strategy. Odd powers? Save one for substitution. Even powers? Use half-angle identities to reduce them.
Strategies for Trig Integrals
Strategy for
Save one factor and use to express remaining factors in terms of sine:
Then substitute .
Save one factor and use to express remaining factors in terms of cosine:
Then substitute .
Use the half-angle identities to reduce the powers:
Also helpful:
Strategy for
Save a factor of and use to express remaining factors in terms of tan:
Then substitute .
Save a factor of and use to express remaining factors in terms of sec:
Then substitute .
Product-to-Sum Identities
For integrals like , , or :
| Product | Sum Identity |
|---|---|
Trig Powers
Visualizing
Worked Example: Odd Sine Power
Strategy (b): Odd Sine Power
Evaluate .
The power of sine is 3 (odd). Use Strategy (b): save one for , convert the rest using .
Level Up Examples
Example A: Odd Cosine Power
Evaluate .
Cosine has power 3 (odd). Use Strategy (a): save one , convert remaining .
Example B: Both Even Powers
Evaluate .
Power of cosine is 2 (even), no sine. Use Strategy (c): half-angle identity .
Example C: Even Secant Power
Evaluate .
Secant has power 4 (even). Use Tan/Sec Strategy (a): save for , convert remaining .
Example D: Odd Tangent Power
Evaluate .
Tangent has power 3 (odd). Use Tan/Sec Strategy (b): save for , convert .
Example E: Product-to-Sum
Evaluate .
This is with different arguments. Use Product-to-Sum: .
Practice Quiz
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