Homework Help: Math - Trig Substitution

Posted by Marissa on Monday, September 8, 2008 at 3:55am.

How can I solve the integral of x^3√(9-x^2) dx using trigonometric substitution? ?
∫ x^3√(9-x^2) dx

So then I know that
x = 3sinθ
dx = 3cosθdθ

When I substitute, it becomes

∫ (3sinθ)^3 * √(9-(3sinθ)^2) * 3cosθdθ

= ∫ (27sin^3θ * (3 - 3sinθ)* 3cosθdθ

Is there any way to furter simplify this before I solve it? And if there isn't, how would I go about solving it?

Math - Trig Substitution - Damon, Monday, September 8, 2008 at 5:19am
∫ x^3√(9-x^2) dx

So then I know that
x = 3sinθ
dx = 3cosθdθ

When I substitute, it becomes

∫ (3sinθ)^3 * √(9-(3sinθ)^2) * 3cosθdθ

BUT √(9-(3sinθ)^2) = 3 √ (1-sin^2)θ
1 - sin^2 = cos^2

= ∫ (27sin^3θ * (3 - 3 COS θ )* 3cosθdθ

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