Verified answer

∫ cos^3 (x/9) dx

You can use triple angle formula

cos(3A) = 4cos^3(A) - 3cos(A)

So

cos^3(x/9) = (1/4) [cos(x/3) + 3cos(x/9)]

Integration of that is

(1/4) [ 3sin(x/3) + 27sin(x/9) ] + C

= (3/4) [ sin(x/3) + 9sin(x/9) ] + C

when x = 3π/2

(3/4) [ sin(x/3) + 9sin(x/9) ]

= (3/4) [ sin(π/2) + 9sin(π/6) ]

= (3/4) [ 1 + 9(√3 /2) ]

= (3/4) + 27√3 /8

Similarly, when x = -3π/2,

(3/4) [ sin(x/3) + 9sin(x/9) ] = -(3/4) - 27√3 /8

Upper limit - Lower limit

= (3/2) + (27√3 /4)

= (6 + 27√3)/4

∫ cos^3( x / 9 ) = (3/4)(9sin(x/9)+sin(x/3) and the definite integral is (3/2)[1+(9/2)sqrt(3)]