How do you find the antiderivative of # cos^4 (x) dx#?

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Answer 1 · 2016-06-28T19:54:55

You want to split it up using trig identities to get nice, easy integrals.

#cos^4(x) = cos^2(x)*cos^2(x)#

We can deal with the #cos^2(x)# easily enough by rearranging the double angle cosine formula.

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

#cos^4(x) = 1/4(1 + 2cos(2x) + cos^2(2x))#

#cos^4(x) = 1/4(1 + 2cos(2x) + 1/2(1 + cos(4x)))#

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

So,

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

#int cos^4(x) dx= 3/8x + 1/4*sin(2x) + 1/32*sin(4x) + C#