Integration of even powers of sine and cosine

Thread starter lydia_y620
Start date Nov 7, 2017
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Calculus Cosine even Integration Sine Trig identities

In summary, the formula for integrating even powers of sine and cosine involves the use of trigonometric identities and results in a combination of sine and cosine terms. Integrating odd powers only requires basic integration techniques and results in only one type of trigonometric function. The power of sine and cosine must be a positive even integer, but other real numbers can be integrated using different techniques. If the integral contains both even and odd powers, it can be split into two separate integrals. Real-world applications of integrating even powers of sine and cosine include solving problems involving periodic motion and using the Fourier series in fields such as signal processing and music theory.

Nov 7, 2017

lydia_y620

Homework Statement

Homework Equations

cos²x = (1+cos2x)/2
sin²x = (1-cos2x)/2

The Attempt at a Solution

I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?

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Nov 7, 2017

PeroK

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What about using ##cos^2x + sin^2x =1##?

Nov 7, 2017

lydia_y620

PeroK said:

What about using ##cos^2x + sin^2x =1##?

okay, I've figured it out. Thanks!

1. What is the formula for integrating even powers of sine and cosine?

The general formula for integrating even powers of sine and cosine is:
∫(sinⁿx) dx = −(1/2)(sinⁿ⁻¹x)cosx + (n−1)/2∫(sinⁿ⁻²x) dx
and
∫(cosⁿx) dx = (1/2)(cosⁿ⁻¹x)sinx + (n−1)/2∫(cosⁿ⁻²x) dx

2. What is the difference between integrating even and odd powers of sine and cosine?

The main difference is that integrating even powers of sine and cosine involves the use of trigonometric identities, while integrating odd powers only requires basic integration techniques. Additionally, the resulting integrals for even powers will have a combination of sine and cosine terms, while odd powers will only have one type of trigonometric function.

3. Can the power of sine and cosine be any real number?

No, the power of sine and cosine must be a positive even integer in order for the integration to be possible using the general formula. However, other real numbers can be integrated using other techniques such as substitution or integration by parts.

4. How do you handle integrals with both even and odd powers of sine and cosine?

If the integral contains both even and odd powers of sine and cosine, it can be split into two separate integrals, one for the even powers and one for the odd powers. Each integral can then be solved using the appropriate formula or technique.

5. Are there any real-world applications of integrating even powers of sine and cosine?

Yes, the integration of even powers of sine and cosine is commonly used in physics and engineering to solve problems involving periodic motion, such as the motion of a pendulum or a spring. It is also used in the Fourier series, which is used to analyze and synthesize periodic functions in fields such as signal processing and music theory.