Integrating (9-x^2)^{5/2}: Solving Integration Frustration

[negatifzeo]

Homework Statement

$\int (9-x^2)^{5/2} dx$

Homework Equations

The Attempt at a Solution

Letting$$x=3sin\theta$$ and $$dx=3 cos \theta$$,

[itex] 243\int(1-sin^2\theta)^{5/2}3 cos \theta d\theta[/tex]

Taking the 3 out gets

$$729 \int (1-sin^2 \theta)^{5/2} cos \theta d\theta$$

I'm not sure where to go from here. Do I combine everything algebraically and then do a u sub? Or do I integrate by parts here?

 

[AEM]

Homework Statement

$\int (9-x^2)^{5/2} dx$

Homework Equations

The Attempt at a Solution

Letting$$x=3sin\theta$$ and $$dx=3 cos \theta$$,

[itex] 243\int(1-sin^2\theta)^{5/2}3 cos \theta d\theta[/tex]

Taking the 3 out gets

$$729 \int (1-sin^2 \theta)^{5/2} cos \theta d\theta$$

I'm not sure where to go from here. Do I combine everything algebraically and then do a u sub? Or do I integrate by parts here?

Well, you could remember that $$(1 - sin^2 \theta) = cos^2 \theta$$ so that your integrand becomes $$cos^6 \theta$$ and then look that integral up in a table of integrals, or look up a reduction formula for $$cos^6 \theta$$. You'll be able to integrate the expression that results from the reduction formula.

 

[gabbagabbahey]

Well, you could remember that $$(1 - sin^2 \theta) = cos^2 \theta$$ so that your integrand becomes $$cos^6 \theta$$ and then look that integral up in a table of integrals, or look up a reduction formula for $$cos^6 \theta$$. You'll be able to integrate the expression that results from the reduction formula.

Careful; $$(\cos^2 \theta)^{5/2}=|\cos^5 \theta|\neq\cos^5 \theta$$ in general.

 

[AEM]

Careful; $$(\cos^2 \theta)^{5/2}=|\cos^5 \theta|\neq\cos^5 \theta$$ in general.

You are, of course, correct. I should have been more careful.