Integral of cos^4(x/4) dx ?

1 Answer
Øko
Feb 26, 2018

#I=1/8(3x+8sin(x/2)+sin(x))#

Explanation:

We want to solve

#I=intcos^4(x/4)dx#

When we have an integral of a trigonmetric function,
to some power, we often want to reduce the power of the trigonometric function, which will make the integration easier

So let's reduce the power of cosine, using

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

Thus

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

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

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

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

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

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

So the integral becomes

#I=1/4int(3/2+2cos(x/2)+1/2cos(x))dx#

#=1/4(3/2x+4sin(x/2)+1/2sin(x))+C#

#=1/8(3x+8sin(x/2)+sin(x))+C#

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