# Integrals of Trig Powers: Sin^2dx and Cos^2dx

• mathguyz
In summary, the conversation discussed the emphasis on multiple powers of trig functions in the calculus book and the difficulty of integrating sin^2dx and cos^2dx. It was suggested to use certain identities to make the integration easier and it was mentioned that these types of integrals are commonly encountered in physics, specifically in dealing with spherical harmonics.
mathguyz
The calculus book places an emphasis on multiple powers of trig functions in the book. Does anyone here really know what the integral of sin^2dx is? What about the integral of cos^2dx is? I don't think I ve actually ever seen it.

You can calculate them rather easily if you use $$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$$ and $$\cos(x) = \frac{1}{2} (e^{ix}+e^{-ix})$$.

mathguyz said:
The calculus book places an emphasis on multiple powers of trig functions in the book. Does anyone here really know what the integral of sin^2dx is? What about the integral of cos^2dx is? I don't think I ve actually ever seen it.

to integrate either cos2x or sin2x with respect to x, the identity:

cos2x=cos2x-sin2x=2cos2x-1=1-2sin2x

Will help

Integrals involving higher powers of trig functions occur all the time in physics, one example of this occurs when dealing with spherical harmonics, which are a set of special functions that occur in quantum mechanics and electromagnetism. If you have ever seen s,p,d,f orbitals in chemistry, know that the shapes they are showing you correspond to spherical harmonics, and that working with these requires you to integrate higher powers of trigonometric functions.

## 1. What is the general formula for integrating trigonometric powers?

The general formula for integrating trigonometric powers, including sin^2(x) and cos^2(x), is: ∫sin^nx dx = -1/n * sin^(n-1)(x) * cos(x) + (n-1)/n * ∫sin^(n-2)(x) dx and ∫cos^nx dx = 1/n * cos^(n-1)(x) * sin(x) + (n-1)/n * ∫cos^(n-2)(x) dx. This formula can be used to integrate any trigonometric power with an odd or even exponent.

## 2. How do you integrate sin^2(x) and cos^2(x)?

To integrate sin^2(x) and cos^2(x), you can use the half-angle formulas: sin^2(x) = (1-cos(2x))/2 and cos^2(x) = (1+cos(2x))/2. These formulas can be substituted into the general formula for integrating trigonometric powers to obtain the final integral.

## 3. Can you use trigonometric substitutions to integrate sin^2(x) and cos^2(x)?

Yes, you can use trigonometric substitutions to integrate sin^2(x) and cos^2(x). For example, for sin^2(x), you can use the substitution u = sin(x) and for cos^2(x), you can use u = cos(x). These substitutions will transform the integral into a simpler form that can be solved using basic integration techniques.

## 4. How do you use integration by parts to integrate sin^2(x) and cos^2(x)?

To use integration by parts to integrate sin^2(x) and cos^2(x), you can choose sin(x) or cos(x) as the first function and the other trigonometric function as the second function. This will result in a reduction formula that can be used to solve the integral iteratively.

## 5. Are there any real-world applications of integrating trigonometric powers?

Yes, integrating trigonometric powers has many real-world applications in fields such as physics, engineering, and statistics. For example, in physics, the integration of sin^2(x) and cos^2(x) can be used to calculate the area under a velocity-time or acceleration-time graph. In engineering, it can be used to calculate the power output of a sine or cosine wave. In statistics, it can be used to calculate the area under a normal distribution curve.

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