The integral (x^3sqrt(9^2-x^2)) can be transformed using a substitution to 59049 integral cos^4(x) -cos^2(x)
i've figured out how to get the two trig functions but cant understand how to write it as a composite function as an answer
thses are the two trig substitutions of x= 9sin(x) and u =cos(x). what is the composite function of those two? please and thank you
if x = 3sinθ
then dx = 3cosθ dθ
and that gives
x^3 √(9-x^2) dx = 27sin^3θ √(9-9sin^2θ) 3cosθ dθ
= 243 sin^3θ cos^2θ dθ
Not sure where you'd get 59049=243^2 as a factor ...
The composite function is:
59049 ∫ cos^4(x) - cos^2(x) dx = 59049 ∫ (9sin(x))^4 - (cos(x))^2 dx