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Integration By Substitution.

Hi,

Could someone please tell me if I have got the correct answer to this question below:-

Use the substitution x = 3Sintheta to show that:

Integration of sqrt(9-x^2) dx = Integration of k cos^2 theta dtheta where the values of the constants a, b, and k are to be found.

Hence evaluate the integration of sqrt(9 - x^2).


Now I really hate this part of integration as I don't see where the next step is most of the time. :frown:

I didn't get to k cos^2 theta dtheta. I got to 3costheta dtheta / 3costheta ending up with 1 dtheta and limits of pi/2 and pi/6. My answer to this integral was therefore pi/2 - pi/6 which I think is wrong.

Could somone please help me out. :frown:
Thanks.
(edited 12 years ago)
Reply 1
Avatar for olipal
olipal
1
Letting u=3sin⁡θu=3 \sin \theta:

du=3cos⁡θ dθdu=3 \cos \theta\ d \theta so,

∫9−x2 dx⇒∫9−9sin⁡2θ.cos⁡θ dθ=3∫cos⁡2θ dθ\displaystyle \int {\sqrt{9-x^2}}\ dx \Rightarrow \displaystyle \int {\sqrt{9-9 \sin^2 \theta}}. \cos \theta\ d \theta = 3 \displaystyle \int {\cos^2 \theta}\ d \theta

Can you see where to go from here?
(edited 12 years ago)
Reply 2
Avatar for FutureMedic
FutureMedic
OP
Original post by olipal
Letting u=3sin⁡θu=3 \sin \theta:

du=3cos⁡θ dθdu=3 \cos \theta\ d \theta so,

∫9−x2 dx⇒∫9−9sin⁡2θ.cos⁡θ dθ=3∫cos⁡2θ dθ\displaystyle \int {\sqrt{9-x^2}}\ dx \Rightarrow \displaystyle \int {\sqrt{9-9 \sin^2 \theta}}. \cos \theta\ d \theta = 3 \displaystyle \int {\cos^2 \theta}\ d \theta

Can you see where to go from here?


Oh I see! From there would I use the double angle formula cos2theta = 2cos^2theta - 1?? :smile:
Thanks!
(edited 12 years ago)
Reply 3
Avatar for olipal
olipal
1
Original post by FutureMedic
Oh I see! From there would I use the double angle formula cos2theta = 2cos^2theta - 1?? :smile:
Thanks!

Yep, the after a bit of rearranging you should get something that's pretty straightforward to integrate. Remember to change your limits too! (although from the OP, the fact you are talking about using π2\frac{\pi}{2} and π6\frac{\pi}{6} would imply you've already remembered!)

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