5

Sin % Jo dx I...

Question

Sin % Jo dx I

sin % Jo dx I



Answers

Evaluate the integral.

$ \displaystyle \int \frac{1 + \sin x}{1 - \sin x}\ dx $

Let's talk about this question. Uh We gotta integrate this. Uh We see that there is a seasonings on top. So if he can substitute cause as something then the cynics dx will be uh derivative. So the agenda is to get a cause in the denominator and we can do that easily because we know that the numerous remains other tests and we know that science square X is nothing but one minus cause square X. And now we can do what we were talking about. The Cossacks is let's call you then. Minor Sin X. Dx is do you? Uh This means that cynics takes will be replaced by minus two. You and denominator. We have one plus one is two. So that's two minus U squared. So that can further Britain has won over you square minus route to square do you? Because Uh two is nothing but route to square. This can be read it. And as using the difference of the prophet square, that's you. Plus route to Times U- Route two times in the U. Uh So we can use partial fraction over here. So one over U plus route two times U minus route to is equal to a over you. Plus route two plus B over U minus route to. So that's going to be one is equal to eight times U minus route two plus B times you plus route. All right, so the first substitute first will put a s route. So this stone will become zero and the other time will become too rude to be. This means that the value of B Is 1/2 2. And if we substitute you as minus route to And the stone will become zero and the other time will become -2. Medical -2 to a. Which means that Is equal to -1/2. So this is these are the values of A. And B. Which we can plug in and we can easily integrate now. So the integration, let's continue the red ones with. The integration will now become one over to route to over you. Plus route to the U. Plus. Uh Sorry this is minus because it is minus one or two and be as one over to route to over Hugh minus route to whether do you say? So it's gonna look like uh one over to root can be taken over from the complete expression. So we elephant -1 over you. Plus route two times D. U. Plus Integration of one over U minus route to D. You. So this will become one over to route to. Uh Disintegration is pretty straightforward. The natural log off you go this route to and this is natural log off U minus route to is the constant of integral sc. So finally let's convince this. Using the properties of the log log log log of a. Over B. There's going to be log of U minus route over you press route to but you is nothing but Cossacks we know that. So it's gonna be We're replacing you by cossacks as well. So that's what we call you. A larger log of cossacks minus route to over cossacks plus route two plus a concert of integral to see which is the final answer. Thank you.

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

Let's evaluate to given integral here we could go ahead and try. U Substitution will take you to be sign of X. Then we have Do you is co signed the X So plugging it into this in a girl, we have a top. That's just do you on the bottom. We have one minus you. Now, here. If this one minus he was bothering you. You could go ahead and do it. Use up. Then you could write. This inner girl is negative. One over. You won over W D w So that's negative. Natural log. Absolute value W plus e. Now we use our substitution to replace w with you. So that's one minus you on the absolute value. And then finally, we use our U substitution or original substitution to replace you with X, and that's your final answer.

So we want to integrate sign X. Cosign X. Dx. Um And the way to go about this is to try to make a U. Substitution. So we want to do a U. Sub. So we need to decide which piece of this we want to be you. Is it going to be the sign of the coastline? And the way you should think about it is this so you know the derivative of sine is co sign. And whenever we use U substitution we always need to take the derivative. So let's let's sign of X. Bu. So let's let you equal sine of X. And then what is do you do you is going to be the derivative of sine which is co sign X. D. X. So um we have the integral of you and then co sign X. D. X. Is D. U. So um if we integrate with respect to you we have you squared over two um Plus some constancy. And then we just have to substitute back in our original um function in terms of X. So we've got sine squared X over two plus C.


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