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Evaluate the integral.

$ \displaystyle \int \frac{x^2}{\sqrt{9 - x^2}}\ dx $

$\frac{1}{2}\left(9 \sin ^{-1}\left(\frac{x}{3}\right)-x \sqrt{9-x^{2}}\right)+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 3

Trigonometric Substitution

Integration Techniques

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Let's evaluate the integral of X squared over the square of nine minus X squared. Let's use a trick substitution here since in the radical tyrannical expression insides of the form a squared minus X squared we'LL use the tricks of X equals a scientific data. In our problem, we have a equals three. So we have X equals three scientists. This is our tricks up. So we have the eggs is three coz I'm data. Also, we have the square root of nine minus six where let's go to simplify this radical first this is nine minus nine sine squared. We could pull off the nine outside the radical that becomes a three one minus sign square on the inside. So the three square root of co sign squared, which is three co sign So this inner world becomes so X squared becomes nine signs where and then DX becomes three coats and data data and in the denominator we just worked that out over here, and that's three co sign so we could cancel these three co signs. So we have nine into girl science. Where here We could use one of the Pythagorean or one of the true identities the double angle formula to rewrite this as one minus coastline of two theater, all divided by two. And then now we could integrate. So we have ninety eight over too, minus nine, signed to theatre number four and then plus our constant C of integration. So at this point, we could use the double ankle formula for sign to rewrite. This is to sign Data Co signed data with ninety eight over too minus nine Over too. Signed data cosign data plus e. So leave evaluated the new girl. But in order to get our answer back in terms of the original variable X, we should draw the triangle. So let's go to the next patient. Do this. So our tricks up was X equals three signed data. So this means Scient Ada is X over three. So let's draw any right triangle that has this property. So here's data and then sign opposite over hypotenuse. So then we can use for that reindeer tto find the remaining side B. So we know B squared. Plus X squared is nine. So that B is equal to the square room of nine minus x work. So now we have all three sides of the triangle, So coming back to our original. So where we left off in the previous page, we had nine over Sue Daito minus nine over too. Signed data, coastline data plus e, not the data itself. We can get from our tricks up. So taking this equation up here, solving for data. So we take Sinan person both sides so we can write that for data. So we have a nine over too Sign in verse, eggs over three, minus nine over, too. And now sign Times Co sign. So sign is exploratory and co sign is be over three. So be is nine minus X squared inside the square root all over three. And then we can go ahead and cancel this nine with the nine in the bottom. And we still have a two in the bottom. And there's a final answer

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