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Evaluate the definite integral: 5/2 sin X dx -J/2 1 +x8...

Question

Evaluate the definite integral: 5/2 sin X dx -J/2 1 +x8

Evaluate the definite integral: 5/2 sin X dx -J/2 1 +x8



Answers

Evaluate the integral.

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

Let's start up by re reading this in a roll. So using the double angle formula for sign and then on the denominator, we could write This is one plus co signed square and then square that And then let's try use a beer, coastlines where and then here was too negative. Do you to sign ex co sign X after used the chamber So we get negative in a girl. So we have our do you and then one plus u squared. So if you want, you could put a one up there. Now you may have memorized this formula and if so, you can just write tan inverse of you. If you forgot this fact, just do it tricks up on the cinder girl you would let you equals tan data and you have to solve that in the role. So here we have our HQ tan of you with a minus out front, plus our constancy and let's go ahead and replace you with you so that we started with. So we have negative Arc tan co science for of X plus our constancy. And that's our final answer

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

This problem is from Chapter seven section to Problem number 17 in the book Calculus Early Transcendental eighth Edition by James Stewart And here we have a indefinite integral of sine squared of X times sine of two x. So first thing we can do is apply a double angle formula to rewrite sign of two X So we have integral sign square eggs and then using our our double angle formula for sign. We have two sin x co sign X, the X so we can pull this to outside the integral and combine these signs to get assigned Cube. And this would be a good time to apply a u substitution. Here we can take you to be cynics so that do you is co sign of X dx. So if we use this use sub are integral becomes two times integral of u cubed, do you? We can evaluate this using the power rule for the integral two times U to the fourth power over four plus c. So here we can back substitute from you back to Synnex using our our u substitution. And this to over four becomes a one half. So we end up with signed to the fourth power of X over to plus C. And there's our answer

Let's start off by just expanding this into Grand X Square to Excite X Science, where then we could go ahead and split this sense of three generals. Oops, think of the ex first and then in a gruel of science squares. So for the first one, just use the power room here. Let's use and enrich my parts. Let's take you to be x two u equals the x t me signed the eggs. So he is negative. Cosign x sound. Recall the formula to be replaced the integral with UV minus in rule be, do you? So we have plus and then two times. So now we do you times V So this times this that's negative. X co sign X and then minus in a rule z Do you? So I have a negative co sign. Let me just go ahead and cancel those minuses. And then finally, for the last integral. It's used a half angle formula for sign, so it's one minus co sign to X, and we pulled out the two. So let's simplify this integral of co sign a sign. Don't forget the two out here. So here, in a goal of one X. Don't forget the one half hour in the front, out and then the end. Roll over here that'LL be a minus sign, too. It's over, too. But then we have another two over here. So as a four plus our constant of integration, see? And that's your final answer.


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