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Evaluate the integral.

$ \displaystyle \int \sqrt{1 - \sin x}\ dx $

$$2 \sqrt{1+\sin x}+C$$

Integration Techniques

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let's start here by taking you use up. Let's take you to just be sign of X, then do you that she's co sign X t X And now the difficulty here will be t just free. Write this in terms of you because we would liketo have d x by itself here. That means we would liketo have something like this, but not quite because yes, we do have the X by itself. But the left side has a u an ex so have to rewrite this in terms of you. So we know that co signs where is one minus science where and then just take a sweet room here and then using our use up. This is one minus, you swear. So let's do do you over one minus you square equals DX. So now rewriting on Integral This is square rule of one minus you up top and then for DX we have do you and then over one minus is where? Oops! Next. Let me just go ahead and simplified that denominator. Who's that? Should not be swearing there as he won mine issue of talk. Then the bottom have one minus you and then one plus you after factoring and then just splitting the radical of over the private. So all I did there was just use the fact that he multiplied two positives in the radical We can write this and then I could go ahead and cross off those terms there er and purchase left over with do you over the square of one. Plus you. Now this is much simpler, integral than what we started with. So for this one, you could do it. Another use of here, Let's go to the next page. This time let's take me to be one plus you So that D v equals to you that we could write this in a girl as one over Rudy Davey. So if we want was to be to the negative one half. So that's to be two the one half. Plus, he go ahead and replace G with you coming from our second use of up here and then recall the original use of on page one was signed X. So now we come back over here and replace that you with sine X, and that's our final answer