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2 (a)+ 1 sin t)9 xdx; cos t dt; cotxsin t...

Question

2 (a)+ 1 sin t)9 xdx; cos t dt; cotxsin t

2 (a) + 1 sin t)9 xdx; cos t dt; cotx sin t



Answers

$\int \frac{(\sin x-\cos x) d x}{(\sin x+\cos x) \sqrt{\sin x \cos x+\sin ^{2} x \cos ^{2} x}}=$ (A) $\operatorname{cosec}^{-1}(1+\sin 2 x)+c$ (B) $-\operatorname{cosec}^{-1}(1+\sin 2 x)+c$ (C) $\sec ^{-1}(1+\sin 2 x)+c$ (D) $-\sec ^{-1}(1+\sin 2 x)+c$

Well, the given integral will make the substitution you. Would you goto one plus scientific So differentiating both sides We have deal You go to nasty duty so they're given integral becomes interesting. Um, do you buy you? Which is photo? I learn you plus C Now it's up shooting back. Well, Ln one plus sign plus c, let's see the constant of integration.

500 MG group maybe half. So let's solve this. Coming up to be integral behalf one where we have athletic this signings over because it's supposed to be for taxes. All six over plan. It's supposed to be at texas 1/4 6 positive one or work find the X. Just on this game coming up to be America we have in the group than one over. We have one plastic Phoenix or robotics. Lasutina one Plus Cortex. Former science just be award here. Now you can find this so we get to be integral cynics politics or we have to be my necks one block planning's all of them. Well I think one blood mr get here. So now we just keep on solving this. We get here fine thanks bullets or I'll be signing books. All of the friends correct plus vortex or the bill false corrects this. Can I do cynics 12 6 dates or we have science students discussed this is one area. So one plus findings plus politics. Were born here. Now what we can do will just land right here. It'll be fine X. Politics or finance plus four six plus one. Anybody right here we have cynics plus cortex negative one over Synnex plus cortex. Everyone here we have beards Integrity of your last command to be signing for six times. Science not giving Science critics for six plus cynics false negative panics process or we have now is A plus B in early this morning is going to the museum. We get here signings plus four six hold good -1 We have beers. Now let's keep on simplifying this. So. Mr. Marketing, integral science politics, politics, positive clinics, course correct negative signings. Both. Or we are not actually scientists prices one. We get here uh to cynics politics and area. Yeah. one over. Do we get from? And we get here is going to be cynics V. It's Impossible for six be it then negative one. Bs. We're just not solved. So we get 1/2. Process will give cynics finance will give negative for six and then we have negative eggs plus the bow. This uh go over to their option Vietnam sinus. And of course it's negative uh Alex over to the same thing. So option B. Is uh thank you.

So we're looking at this function and were asked for the anti derivative, um, sign of tea, plus the hyperbolic hopes two times the hyperbolic sign of teeth. And as I'm looking at the anti derivative, what I would do is just think about well, I know the derivative of scientists. Cosine. I know the derivative of co sign is negative sign. So let's fix that by making this negative. So the derivative of negative cosign would make it positive sign. Now, what's nice about the hyperbolic sign and hyperbolic co sign is they're both positive, no matter what. So the anti derivative of the hyperbolic sign is the hyperbolic cosine of tea. And then don't forget about your plus C because if I asked you for the derivative of this kind of already explained it this way, um, well, the derivative of some constant would be plus zero, Andi. That's why we have to write this down. Because there could be some value shift. The graph up and down, um, has the same derivative. But you don't have to write plus zero there. But this does confirm that, uh, you know, my answer that circled in green is correct. Right there

So let's start by showing that C N X is equivalent to the expression post Dynex, all divided by one minus sine squared. Let's and let's recall the Magritte identity, that state that states that post science where that's science weird X equals one. As a result, we know that CO sign square debts is equivalent to one minus sine squared X. So if we work with the right side, we can show that coastline relax, but it by one minus sine squared X Is that what went to co sign? Abets if I did I well, since 11 the signs for that's is equivalent to close sine squared X. We know that close I next about it. Coastlines scored axes, a little expression reducing we now obtained, one divided by coastline of X, and the reciprocal of coastline of X is are seeking X function. Next, less integrate seeking X with respect to X, using this identity that we just proved so seeking X is now equal. Want to co sign events divided by the quarter T one minus son square dance. With respect to X, we can now use a new substitution. What's that you represent? Sign of its If you a sign of X, they don't know that deep you is co sign events. Yes. So you now have the integral of, well, coastline of vets terms. The eggs now becomes. Do you? When you have a one in the numerator at one minus science where that's now becomes one minus you squared. So since you will sign about, we want to make sure we swear that you that's what, in fact, er the denominator. Since we have a difference of squares, the denominator factors as one plus you times expression one minus you would expect to you. We can now use portion fraction decomposition to integrate this expression. So I'm setting the bar. Portia Fraction decomposition. We have one divided by one plus you times one minus you. These are both long repeated linear factors so we can place constants over beach, not living each non repeated. When you're factor, multiply both sides by the denominator, which would be one plus you times one minus you, and that will clear the denominator from the left and from the right slowly obtained the equation. One equals eight times expression one minus you, plus B times expression one plus now have some options. We can truly multiply out the right side and compare corresponding coefficients to come up with a system of equations to solve. Or we could substitute values for you that will eliminate certain terms and go on and sulfur and be which I will do in this case. So let's start by letting you equal one. If you was one in our equation, we'll obtain the following one. Minus one would then be serum. So would have one equals two times the or, in other words, be equals 1/2. Next, let's let u equal negative. One sense view equals negative one will make the second term is, he wrote, one minus a negative one would give us a positive, too. So I have one equals two a or in other words, a equals that half Israel so that now out or partial fractions would become 1/2 divided by expression one plus you and 1/2 divided by the expression 19 issue. So we now have the integral because scripts up institute of and intra grows, factoring out the constant of 1/2 times the integral of 1/1. Plus you with respect to you plus 1/2 times the integral of wine divided by one minus you respect to you. Next, we can do substitution zones for the denominator or from experience. From now enjoying a few u substitution problems, we should be able to quickly see that the first integral will now be 1/2 times the natural other the absolute value of one plus you, since we don't get any additional factors from the derivative of one plus year, but maybe for the second when we want to go ahead and do the formal substitution cells to a depue substitution by letting w equal one minus you therefore making DW equal to negative D view or negative DW or negative dw Equalling, do you? It's as a result, we now have for the second extra broke 1/2 times the interval will for did you were going to substitute in and negative feed of you. So it actually took out right that as a negative one times did of you and then the denominator one minus you becomes just a w. So again, this second integral is very similar to how we would do the first If we did say our substitution for the denominator r equals one plus you and the first integral, the irrevocable one. Plus you just give us your d r D You will be be are giving us no additional factor. So you just integrate one over our with respect to our or the natural log of absolute value of our and back substitute to give us a natural above absolute value one plus year. So they were similar to what we're doing for the second interval. So contained right now way have 1/2 times the natural log of the absolute value of one plus U minus 1/2 times the natural off of the absolute value. W also constant. That's substituting in expression for W and then back, substituting our expression for you so we can recall from above that are you waas sign of events. So again recalling that you will sign of that we now have 1/2 times the natural log. But that's the value of one plus are you expression which was now sign from Epps. Close that absolute value minus 1/2 natural aka the absolute value of one minus her view again will sign of steps close that absolute value science, plus a constant just this one option for the answer. But based on trying to come up with a particular form of the answer, we want to rewrite this expression, using how triple factoring as well as properties of water. So first factoring out a 1/2 and that next over on the side, we have recalled the longer than property that says The natural Have You Divided by V is equal into the natural give you minus the natural movie, and that works in anger border. So in this case that we think about the natural laws of one plus sign. It's where the one plus I nets is representing a your expression and one minus sign of exit is representing like of the expression. We would now have 1/2 times the natural walk of absolute value of one plus sign of ETS provided by the expression one minus sign events close an absolute value, sighing, plus our constant of integration, which is the answer we were trying to obtain in the end


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