How do you integrate \int \frac{\cos(4x)}{\cos(x)}dx I tried using trigonometric formulas

Ethen Wong

Ethen Wong

Answered question

2022-01-29

How do you integrate cos(4x)cos(x)dx
I tried using trigonometric formulas for turning it into 2cos2(2x)cos(x)dx1cos(x)dx and can solve the second one, but still no idea of how to proceed with cos2(2x)cos(x)dx.

Answer & Explanation

sainareon2

sainareon2

Beginner2022-01-30Added 10 answers

I would suggest that you instead use the cosine’s quadruple-angle formula (which can be derived by applying the double-angle formula twice):
cos(4x)=8cos4(x)8cos2(x)+1
which would turn your integral into
cos(4x)cos(x)dx=8cos3(x)dx8cos(x)dx+sec(x)dx
and you probably know how to evaluate each of these integrals.
Rohan Mercado

Rohan Mercado

Beginner2022-01-31Added 10 answers

Note that
cos2(2x)=(cos2(x)sin2(x))2=(2cos2(x)1)2=4cos4(x)4cos2(x)+1
and that therefore
cos2(2x)cos(x)dx=4cos3(x)dx4cos(x)dx+1cos(x)dx

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