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

Lettingx=3sin\theta and dx=3 cos \theta,

243\int(1-sin^2\theta)^{5/2}3 cos \theta d\theta[/tex]<br /> <br /> Taking the 3 out gets<br /> <br /> 729 \int (1-sin^2 \theta)^{5/2} cos \theta d\theta<br /> <br /> I&#039;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

Lettingx=3sin\theta and dx=3 cos \theta,

243\int(1-sin^2\theta)^{5/2}3 cos \theta d\theta[/tex]<br /> <br /> Taking the 3 out gets<br /> <br /> 729 \int (1-sin^2 \theta)^{5/2} cos \theta d\theta<br /> <br /> I&#039;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?

<br /> <br /> 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&#039;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.