Asked by Marissa
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?
∫ 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?
Answers
Answered by
Damon
∫ 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
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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