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JH
Numerade Educator

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Problem 33 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \sqrt{3 - 2x - x^2}\ dx $

Answer

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

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Video Transcript

for this in a girl. Let's go ahead and start off by completing the square inside the radical. So first, let's write this in and I'm getting this by taking the original expression inside the square root factor of the minus sign from the terms that have X completed this queer here and then simplified. And so now this is what's going with the radical. And so here you can go ahead and do it you so from. So let me go and rewrite this as to square mine issues were. And then from here we can go ahead and do tricks of But sell us, plug this in and we're getting four minus and then four science, where and then do you? So once we take out a four here So let's just look at this term for one moment. Just simplified this on me. So here, let's fact throw that for you. Get one minus sine squared. So that's it. Square for square Rouco sense where all right, and then you to cause the data. And then we have another one over here, Another to co sign So we'LL have for in a girl called sine squared. Let's use the half angle here for co sign, then in and rate this. And then we have to signed two things now and then divided by two plus e. We could cross off those who's there. And then you could rewrite this term here by using the double angle formula. So now we goes who are original trips up. Do it was to scientific data. So this mean signed equals you over too. Straw that triangle from this? No, there's our data. You divided by two. So buy a bit of a great hero, You could find the other side. And then now you could find sign and call sign. So first two times data get there by taking side Adverse on both sides here. You know where to. So too arc sine you over too. And then two. And now we have signed. Which is you over too cosa which is the radical over too. So here we have one more step and that's to use the fact that we did this You some u equals X plus one. So we'LL go ahead and use this to replace all the use with in terms of X and we could cross off those twos right there. So for our final answer, replacing you. So there's a plus one, and then you becomes X plus one again right there. Then you have your radical, which you could rewrite the radical if you want, and then plus he and there's a final answer.