뒤로Techniques of Integration: Trigonometric Integrals
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Trigonometric Integrals
Integrals of Powers of Sine and Cosine
Integrals involving powers of sin(x) and cos(x) are common in calculus. The strategy for evaluating these integrals depends on whether the exponents are even or odd.
Key Identities:
Double Angle for Cosine:
Double Angle for Sine:
Pythagorean Identity:
Case 1: One Power is Odd
Split off one factor from the odd power to form the differential (du) for substitution.
Rewrite the remaining even power using the Pythagorean identity.
Let (if sine is odd) or (if cosine is odd).
Example 1:
Since the power of sine is odd, write .
Convert to .
Let , .
Integral becomes .
Expand and integrate term by term.
Example 2:
Both powers are even; use double angle identities to reduce the powers.
Case 2: Both Powers Even
Use double angle identities to reduce the powers until the integral is manageable.
Example 3:
Apply the identity:
Integrate:
Result:
Summary Table: Strategies for Integrals of
Case | Strategy |
|---|---|
At least one power is odd | Split off a factor from the odd power, use substitution, and rewrite remaining even power using identities. |
Both powers even | Use double angle identities to reduce powers. |
Integrals of Powers of Secant and Tangent
Integrals involving powers of sec(x) and tan(x) require different strategies depending on the parity of the exponents.
Key Identities:
Case 1: Power of Secant is Even
Split off to form the derivative of .
Rewrite remaining powers of secant in terms of tangent using .
Let , .
Example:
Write as .
Let , .
Integral becomes .
Result:
Case 2: Power of Tangent is Odd and Power of Secant is Positive
Split off to form the derivative of .
Rewrite remaining powers of tangent in terms of secant using .
Let , .
Case 3: Power of Tangent is Even and Power of Secant is Odd or Zero
Rewrite the integrand as a sum of powers of secant.
Use integration by parts (IBP) to derive reduction formulas.
Example:
Let , .
Then , .
Apply IBP:
Use to simplify.
Result:
Summary Table: Strategies for Integrals of
Case | Strategy |
|---|---|
Power of tan is odd, sec positive | Split off , let |
Power of sec is even | Split off , let |
Power of tan even, sec odd or zero | Rewrite as sum of powers of sec, use IBP for reduction |
General Strategies for Trigonometric Integrals
When integrating products of powers of sine and cosine, check the parity of the exponents to determine the best substitution or identity to use.
For secant and tangent, use the derivative relationships and Pythagorean identities to simplify the integrand before integrating.
Double angle and reduction formulas are essential tools for reducing the powers of trigonometric functions in integrals.
Additional info: In some cases, if the remaining power after substitution is odd, the integral may require a radical or cannot be expressed in terms of elementary functions. Always check the parity of the exponents before choosing a strategy.
