How to find the antiderivative of cos^4(x) dx?

Landyn Allison

Landyn Allison

Answered question

2023-03-22

How to find the antiderivative of cos 4 ( x ) d x ?

Answer & Explanation

iistiidgllj

iistiidgllj

Beginner2023-03-23Added 6 answers

cos 4 ( x ) = cos 2 ( x ) cos 2 ( x )
We can handle the cos 2 ( x ) easily enough by rearranging the double angle cosine formula.
cos 4 ( x ) = 1 2 ( 1 + cos ( 2 x ) ) 1 2 ( 1 + cos ( 2 x ) )
cos 4 ( x ) = 1 4 ( 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) )
cos 4 ( x ) = 1 4 ( 1 + 2 cos ( 2 x ) + 1 2 ( 1 + cos ( 4 x ) ) )
cos 4 ( x ) = 3 8 + 1 2 cos ( 2 x ) + 1 8 cos ( 4 x )
So,
cos 4 ( x ) d x = 3 8 d x + 1 2 cos ( 2 x ) d x + 1 8 cos ( 4 x ) d x
cos 4 ( x ) d x = 3 8 x + 1 4 sin ( 2 x ) + 1 32 sin ( 4 x ) + C

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