Evaluate the integral.

[etobias]

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.

[Integral] x^3 * [square root](9-x^2)dx; x = 3 sin [theta]

 

[Gene]

Are you sure the integral isn't from 0 to 3?

x=3sin(T)
dx=3cos(T)dT

x^3=27sin^3(T) = 27(1-cos^2(T))sin(T)

sqrt(9-x^2) =
sqrt(9-9sin(T)^2) =
sqre(9(1-sin^2(T))) =
sqrt(9cos^2(T)) =
+3cos(T)

x^3*sqrt(9-x^2) =
{27(1-cos^2(T))sin(T)}*{+3cos(T)}
Can you take it from there?