How to Integrate cos4 x Using the Double Angle Formula

In summary, the integral of cos4 xdx can be found by using the identity cos4x = ((cos2x)2 + 2cos2x + 1)/4 and simplifying it to ((cos4x + 1)/2 + 2cos2x + 1)/4. This can then be split into four integrals, resulting in the final answer of (sin4x)/32 + (sin2x)/4 + (3x)/8 + C.
  • #1
gerry7
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0

Homework Statement



Find the integral of cos4 xdx

Homework Equations



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

The Attempt at a Solution



I tried using cos4x = cos2x * cos2x and i simplified it to
cos4x = ((cos2x)2 + 2cos2x + 1)/4

I'm not sure if this is right. How do i go about getting the integral of this expression if it is correct. Thanks
 
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  • #2
So far, so good. You just need to take it one step farther, by replacing (cos2x)2 with (cos4x + 1)/2. At that point you'll have an integral with four terms, and you can split this up into four integrals, each of which is pretty easy.
 
  • #3
Hey i got the answer (sin4x)/32 + (sin2x)/4 + (3x)/8 + C which seems to be right. Thanks for your help :)
 

1. What is a trigonometric integral?

A trigonometric integral is an integral that contains trigonometric functions, such as sine, cosine, and tangent.

2. How do you solve a trigonometric integral?

To solve a trigonometric integral, you can use various techniques such as trigonometric identities, substitution, and integration by parts.

3. What are the common trigonometric identities used in solving trigonometric integrals?

Some common trigonometric identities used in solving trigonometric integrals include the Pythagorean identities, double-angle identities, and half-angle identities.

4. Can you provide an example of solving a trigonometric integral?

Yes, for example, to solve the integral of cos^2x dx, you can use the double-angle identity cos^2x = 1/2(1+cos2x) and then integrate each term separately.

5. What are the applications of trigonometric integrals?

Trigonometric integrals have various applications in physics, engineering, and other fields where there are calculations involving periodic functions. They are also used in solving differential equations and in Fourier analysis.

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