Integrating cos^6(x) using cos^4θ - sin^4θ and cos^4θ + sin^4θ

In summary: You will get an expression that has cos(4θ) and cos^2(θ) as factors. From there, use the double angle formula for cos(4θ) and manipulate the expression until you get cos^6(θ) on one side. Then you can substitute back in for cos(4θ) and integrate. In summary, solving this integral requires using integration by parts twice and manipulating the resulting expression to get an expression with only cos^6(θ) on one side. This can be done by keeping cos(4θ) as it is and using the double angle formula for cos(4θ) to manipulate the expression. Once you have an expression with only cos^6(θ) on one side, you
  • #1
synkk
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Using [itex] cos^4\theta - sin^4\theta = cos2\theta [/itex] and [itex] cos^4\theta + sin^4\theta = 1 - \frac{1}{2}sin^22\theta [/itex]

evaluate:

(i) [itex]\displaystyle \int_0^{\frac{\pi}{2}} cos^4\theta \ d\theta [/itex]

adding the two identities given I get: [itex] 2cos^4\theta = cos2\theta + 1 - \frac{1}{4} + \frac{1}{4} cos4\theta [/itex] [itex] cos^4\theta = \frac{1}{8}\left (4cos2\theta + 3 + cos4\theta \right ) [/itex]

integrating this I get the correct answer

(ii) [itex]\displaystyle \int_0^{\frac{\pi}{2}} cos^6\theta \ d\theta [/itex]

from the (i) [itex] cos^4\theta = \frac{1}{8}\left( 4cos2\theta + 3 + cos4\theta \right ) [/itex]

so [itex] cos^6\theta = \frac{1}{8}(4cos^2\theta cos(2\theta) +3cos^2\theta + cos^2\theta cos4\theta) [/itex]

using the double angle formula I simplify down to:
[itex] cos^6\theta = \left(4\left( \dfrac{cos2\theta + 1}{2} \right)cos2\theta + 3\left(\dfrac{cos2\theta + 1}{2} \right) + cos^2\theta cos4\theta \right) [/itex]
[itex] cos^6\theta = \left(cos4\theta + 1 + \frac{7}{2} cos2\theta + \frac{3}{2} + cos^2\theta cos4\theta \right ) [/itex]
considering [itex] cos^2\theta cos4\theta [/itex]

[itex] = cos^2\theta(2(2cos^2\theta - 1)^2 - 1) = 8cos^6\theta - 8 cos^4\theta + cos^2\theta [/itex]

so:
[itex] cos^6\theta = \frac{1}{8}\left(cos4\theta + \frac{5}{2} + \frac{7}{2}cos2\theta + 8cos^6\theta - 8 cos^4\theta + cos^2\theta \right) [/itex]

then the [itex] cos^6\theta [/itex] cancel out :S, any help?

(I've solved this problem using the reduction formula and de moivres theorem but I don't see where I'm going wrong here)
 
Last edited:
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  • #2
You can use integration by parts twice. You will get cos^6 on both sides there, too, but they don't cancel.
Alternatively, keep cos(4θ) as it is and play around with cos^2(θ).
 

1. What is the general formula for integrating cos^6(x)?

The general formula for integrating cos^6(x) is ∫cos^6(x) dx = (1/6)(3cos(2x)+cos(4x))+C.

2. How can I simplify cos^6(x) before integrating?

One way to simplify cos^6(x) before integrating is by using the trigonometric identity cos^2(x) = (1+cos(2x))/2. This can be applied twice to cos^6(x) to get (1+cos(2x))^3/8. From there, expanding and simplifying will give you the final integral.

3. Is there a substitution method for integrating cos^6(x)?

Yes, there is a substitution method for integrating cos^6(x). You can use the substitution u = sin(x) to transform the integral into one involving only powers of sin(x). From there, you can use the power-reducing formula sin^2(x) = (1-cos(2x))/2 to reduce the power of sin(x) and integrate.

4. Can I use integration by parts for cos^6(x)?

Yes, you can use integration by parts for cos^6(x). One possible way is to let u = cos^5(x) and dv = cos(x) dx, which will result in an integral involving cos^4(x). Then, you can use integration by parts again with the same u and dv to eventually get an integral involving only cos(x), which is easy to integrate.

5. Does cos^6(x) have any real-world applications?

Cos^6(x) has many real-world applications, particularly in physics and engineering. For example, it can be used to model the oscillations of a pendulum or the vibrations of a mass on a spring. It can also be used in signal processing to analyze and filter out noise from a signal. Additionally, it has applications in electrical engineering for calculating the power output of an alternating current circuit.

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