How can I integrate (cos(x))^4 using trigonometric identities?

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SUMMARY

The integration of (cos(x))^4 can be effectively achieved using trigonometric identities. By applying the identity cos^2(x) = (1 + cos(2x))/2, (cos(x))^4 can be rewritten as (3 + 4cos(2x) + cos(4x))/8. This transformation simplifies the integration process significantly. The final integral can be computed using standard integration techniques.

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  • Familiarity with integration techniques in calculus.
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I want to integrate (cos(x))^4, but I can't find a formula for (cos(x))^n in my collection.

What is the way to integrate this?
 
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Here's one way
Use the identity cos^2(x)=(1+cos(2x))/2

Then

cos^4(x)=cos^2(x)*cos^2(x) = (1+2cos(2x)+cos^2(2x))/4
=(1+2cos(2x)+1/2+1/2cos(4x))/4
=(3+4cos(2x)+cos(4x))/8
 

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