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Compute the definite integral $$\int_0^1 \left(2-x^2\right)^{3/2}dx$$

Please help me, i suppose $x=\sqrt{2} \sin \theta$ but I just couldn't get the answer.

Thank you

gt6989b
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Kizaru
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  • What happened when using your substitution? – coffeemath Nov 29 '18 at 16:21
  • Using that substitution, what does $dx$ need to be? $dx = \sqrt 2 \cos \theta$ – amWhy Nov 29 '18 at 16:23
  • If you don't tell us where you get stuck, you won't get the most helpful of answers. – amWhy Nov 29 '18 at 16:33
  • https://imgur.com/a/wZSjmZS this is my first try and couldn't figure out where going wrong .i used another way and got the answer. I wanted to know why this is wrong – Kizaru Nov 29 '18 at 16:45
  • Note: I used Wallis in the process – Kizaru Nov 29 '18 at 16:47
  • @Kizaru: In the second line, you correctly have the upper bound $\pi/4$. In the subsequent integrals, you've changed the upper bound to $\pi/2$. – Clayton Nov 29 '18 at 17:45
  • Actually, that is exactly why you're process fails, Kizaru. You can't apply the Wallis formula because of the bounds. – Clayton Nov 29 '18 at 17:50

1 Answers1

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So if $x = \sqrt{2} \sin t$ then $2-x^2 = 2 \cos^2 t$, and also $dx = \sqrt{2} \cos t dt$ and you get $$ \int_0^1 \left(2-x^2\right)^{3/2}dx = 2^{3/2} \sqrt{2} \int_0^{\pi/4} \cos^4 t dt = 4\int_0^{\pi/4} \cos^4 t dt $$ Can you complete this? (Hint: use the relation between $\cos 2x$ and $\cos^2 x$ twice)

(You can see here for how to do it if in doubt)

gt6989b
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  • That all falls out from the mere substitution, and if the asker was stuck on the dx conversion, that was covered. Chances are you stopped where it gets difficult. – amWhy Nov 29 '18 at 16:27
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    @amWhy interesting, will update -- thought he made a mistake in substituting -- but it's a standard technique from here and on, tabulated integral really – gt6989b Nov 29 '18 at 16:29
  • https://imgur.com/a/wZSjmZS. This is my original try – Kizaru Nov 29 '18 at 16:31
  • @amWhy updated with the link to stackexchange post how to solve the trig integral – gt6989b Nov 29 '18 at 16:32
  • I used Wallis in the process – Kizaru Nov 29 '18 at 16:32