Pre calculus HW

Question

Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4.

cos^6(x)

Answer 1

cos^6(x)

=(cos^2(x)(cos^2x)(cos^2x)

=cosxcosxcosxcosxcosxcosx is a product of 6 cosines of x in linear form

or

= (cos2x +1)/2)^3= 0.125(cos2x+1)^3

=(1/8)(cos2x+1)(cos2x+1)(cos2x+1) which is a product of 3 cosines of 2x, each factor in linear form

or with I=Integral

I(cos^6(x))dx = (cos(x)^5(sinx))/6 +5x/16 +5sin(2x)/32 + (5cos(x)^3(sinx))/24 + C

using consine reduction formula

Icos^6(x)dx= (cos^5)(sinx)/6 + 5(.25cos^3(x)sinx+.75I.5cosxsinx + C