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Numerade Educator



Problem 16 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\sqrt{2}}{2}} \frac{x^2}{\sqrt{1 - x^2}}\ dx $


$\int_{0}^{\sqrt{2} / 2} \frac{x^{2}}{\sqrt{1-x^{2}}} d x=\frac{\pi}{8}-\frac{1}{4}$


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Video Transcript

Let's use a trick substitution for this integral because the denominator inside the radical is of the form a squared, minus X squared. I would go ahead and use X equal scientific era, then DX. It's because I'm dead so we can rewrite this integral. And we do have a definite on girl here, So we do have the opportunity or not necessary. It's not required, but it can simplify your work to rewrite these limits of integration in terms of fatal if possible. So here plug in X equals zero and then recall when we do it. Troops up of this form signed data you require. That data is in between negative, however too entirely too. So the only solution to this equation and dishonorable up here is state a equal zero. So our new lower limit will still be zero and then plug in the upper limit for X in. The only solution in this interval that makes this possible is PIRA for so that's our new upper limit. And then we see X Square up here in the numerator, so that becomes science squared after the substitution and then we have DX. So that's co sign D data. And then in the denominator, we have one minus X squared inside. The radical selections come to the side to evaluate this and then use the protector in Toronto for sign and co sign to write. This is co sign squared, and then you can evaluate. This is just cosign data. So that's your denominator. And then we could go ahead and cross off those one half. So here we're left with just pie before and then we have just signed squared only. So here I would use tohave angle formula for science. Queer. That's one minus co sign to date over, too. And then here I'm just pulling out the one half and then I have one minus course I into data left over. So that's one half. Now it's in a very that's data minus scientist eight over, too zero pyro for. And then let's just go ahead and plug those and points in so plugging Piper for first. So we have a pilot or for minus sign pi over too, over two, and then when we plug in zero both terms or zero, we have zero minus sign zero over to, so there's nothing to subtract and then sign a pie over to is just one. So we just have pav rate. So we write this pie over eight and then minus one over four, and this is after we did distribute the one half, and that's your final answer.