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enter image description hereHow do we solve this problem?

$$ \int_{0}^{1/2}\frac{x}{\sqrt{1-x^2}}\, dx$$

All I know is that i need to get the bottom part to equal $\sin^{-1} x$.

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A $u$-substitution will suffice. Let $u=1-x^2$. –  David Mitra Feb 13 at 21:15
    
I have tried it but i get -2 over root U du –  chrissy kwon Feb 13 at 21:16
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Write the indefinite integral as ${-1\over2}\int u^{-1/2}\,du$. –  David Mitra Feb 13 at 21:18
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If you really want to, you can let $x=\sin t$. Then $dx=\cos t\,dt$. –  André Nicolas Feb 13 at 21:21
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@chrissykwon: Look at your sixth line; you've left out a negative sign. So that should be a $-\frac{1}{2}$ to the left of your margin. Now look at the seventh line; what would happen if you differentiated $-\frac{1}{2}u^{1/2}$? You wouldn't end up with $u^{-1/2}$. (This, by the way, is a way to make sure you've integrated correctly: Differentiate your answer and see if you get your question back.) Fix those two spots and let me know what you get. –  dmk Feb 14 at 2:23

2 Answers 2

up vote 4 down vote accepted

If $u = 1 - x^2$, then $du = -2x dx \iff \frac{1}{2}du = x dx$. We can put the limits of integration aside for a moment and rewrite the integral as

$$-\frac{1}{2}\int \frac{du}{\sqrt u} = -\frac{1}{2}\int u^{-1/2} du$$

This integral probably looks like something you know how to handle. So — find the antiderivative and substitute $1 - x^2$ back in wherever you see $u$. Then you can plug in your original limits of integration for the answer.

What do you get?

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thank you for your hep !:) –  chrissy kwon Feb 13 at 21:37
    
the answer says im supposed to get 2- root3 over 2 –  chrissy kwon Feb 13 at 21:38
    
@chrissykwon: Hopefully, that answer's right! What did you get? :) Were you able to find the antiderivative? My book calls what you'd use the general power rule. –  dmk Feb 13 at 21:47
    
Hello, I keep getting (2- root2) over 8 :( –  chrissy kwon Feb 14 at 1:53
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@chrissykwon: If you post an addendum to your question, I'll try to figure out what happened. Please include what you got for the antiderivative using the $u$-substitution. –  dmk Feb 14 at 2:08

Hint: $$ \int_0^{1/2} \frac{1}{\sqrt{1-x^2}}\Big( x\,dx \Big). $$ This should hint at what substitution you should use. If you don't understand this kind of hint, then that's the main thing you need to learn about integration by substitution.

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