evaluate-the-following-integrals-int-fracd-theta1sin-theta

Question

Evaluate the following integrals.
$$\int \frac{d 	heta}{1+\sin 	heta}$$

Ramzi Deek · Accepted Answer

So we have integral of D theta over one plus signed Fada. Well, when you look at this here, I immediately think Okay, let me multiply the top on bottom by one minus. Sign theater one minus. Sign fatal Here. So this will give us integral off. Uh, one minus sign. Fada over one, minus sine squared. Veda The Fed! Now we can split this. So that&#x27;s 1/1 minus sign squared. Seita minus sign theater over one minus sine squared. So this is also remember one minus sine squared Theta is co sine squared things. So that&#x27;s one over co signed squared theta. And this is minus sine theta over co signed Squared fada. Uh, so one over co sine squared is also second squared, Seda. Now I&#x27;m gonna doing integral here, So sign over Coastlines Quarter. That&#x27;s also signed over co sign which it stand Data times one over co sign. So that&#x27;s also seeking Fada. So now we have two standard into girls here. So the integral off seeking squared is tangent Fada minus the integral off tangent. Ah, seeking is just second plus c. So this is your final answer here

Robert Daugherty · Answer

Section seven. Not one problem. 40. This is the introduction to integration techniques here we&#x27;re dealing with intervals will have to do some kind of algebraic or Trigana metric manipulation of the Inter Grant to get into a form where I can recognize anti derivatives. So here, listen, sort of rewrite ISS as one of the common things is, you know, a lot more about signing co sign than you do other trick function. So let&#x27;s rewrite this in terms of the sine function. And then I&#x27;d say Okay, well, now let&#x27;s just get a common denominator in the bottom of this fraction. And that gives me signed a to minus one over sign data data. And so this is the integral Scient, Ada or scientific A minus one di fada. Now I can multiply everything by scientist a plus one of our Sunday two plus one. So here I would have science data scientist Adam minus one. And then let&#x27;s just multiply by one in the form of scientific plus one over scientific a plus one And then what? I end up within the top of this fraction I get sine squared of theta no sign of data and in the bottom. I have signed squared of data minus one d theta. Now you know that the sine squared of data So coastline square that they that is equal to one. So science square, that data minus one is minus co sine squared. So that&#x27;s going to simple by the bottom of this denominator. So that gives me sine squared data science Taylor over minus co sine squared data data. And so now what I could I could break this into two separate integral. So this is going to be, um, minus thean a role sine squared data Arcos sine squared, Dana and then do you data And then, minus the interval of scientific data over coast sine squared date, I d think so. In that first in a row, sign over co sign that is tangents, that this is mine. It&#x27;s the integral change. It squared data minus the integral um, sine theta over coastline squared Data di fada. In this second integral, I can let you equal the co sign of Fada. Then d&#x27;you is minus science data de theater in the first integral Tangent Square iss seeking square minus one So I can write. This is the integral of secret squared of data Monets one de Dana. And then when you see a minus in front of that sign so that minus scientific data, that is, do you So this is just the integral off one of our U squared. Do you now? The anti derivative seeking squared, that is the tangent said this gives me minus tended of data and then the integral of minus one. That&#x27;s minus state of it with a negative sign of right, that&#x27;s just gonna be plus data and then one over you square the interval that is gonna be minus one over you, plus a constant of integration. What was our substitution for you? You is the co sign of data. So what I get here ISS minus change it of data plus data and then one over the co sign. That&#x27;s just the seeking of data, plus a constant of integration. So here is an answer. Um, if your answer was in terms of signing co sign so you can see with these values that their equivalent forms this is minus science data over co sign data plus state on minus one over coastline data, plus a constant of integration so I could see another form of this would be minus one plus scientific data overcoats. And they did, plus data, plus a constant of integration. So another form of this, um, integration.

Robert Daugherty · Answer

Section 67 number 14 were dealing with the integration by parts, which is the formula. If I can recognize an in a row in the form of the in a row of you Devi, I can integrate by parts with the formula you ve minus integral of VD You so basically, if I can find it in the form of you TV but it&#x27;s easier to integrate VD you then I&#x27;m in business. So typically you will let you if you have a polynomial times I&#x27;ve got a polynomial on a trick function Let you be the polynomial function in most cases because d you will be equal to d theta So you can see that when d&#x27;you appears in that second interval that&#x27;s gonna be a a simplification of that interval. So now let&#x27;s just see can we integrate easily? The the other parts of Devi is equal to sign of fate a plus one d theta then V would be the Inderal that so when they integrate, that&#x27;s minus Cosan of data plus one. So that tells me that this expression is going to be you ve so that&#x27;s minus data plus one co sign of data plus one minus the inner grove. Edie, You, Which is gonna be plus the integral with co sign of fate a plus one d theta and I can integrate the cosign really rather easily. So this final answer become to minus data plus one co sign data plus one. And then when you generate the co sign, you&#x27;re gonna get plus the sign of data plus one plus.

J Hardin · Answer

here. Let&#x27;s good and multiply by one minus co sign and then we have one minus co sign and then here on the bottom. You should get one minus co sign squared, and that&#x27;s equal to science work. So from there, let&#x27;s just split this into two. Look, Quinn, and then we can go ahead and rewrite thes and liberals. There&#x27;s something we recognize, so that&#x27;s Kosi can square. And then here I write this as coasts and over sign and then one over sign. So that becomes Cose Engine because he can. And then here we have. We know these for the table, great and your constancy, and that&#x27;s our answer.

R M · Answer

problem number 40 the figuration and ah, sign. I think there leave later Over square root of one plus co sign. Ah, we&#x27;re fate. So go sign. Faith is equal to you and negative sewn plead The the faith is equal to be new It&#x27;s been sewing state. Do you say that is equal to negative? Do you? So the integration is equal to negative the u over square root of one plus you square he is equal to negative the integration off Do you over square root off one plus you square is equal through in anything showing members Oh, you plus C which is equal through a negative showing angers for a cool design taint of glossy

Foster Wisusik · Answer

Okay, This question wants us to integrate this function. So, like, usual, let&#x27;s choose our U substitution to be equal to the angle inside our trick function. Because this tells us that d&#x27;you would be equal to negative one over theta squared d theta and we have a positive one over theta squared inner integral. So let&#x27;s multiply both sides events by negative one to get negative. Do you equals one over theta squared d theta. So now we can plug in and get the integral of signed you co sign you times negative to you or negatives integral of sign You Coast cite you, Did you? So starting at this point, we can either use a trick identity to condense this into one function or if you don&#x27;t know that off the top of your head, we can actually do another substitution. So this time let V be equal to sign you. Which means that D V is equal to co sign of you. Do you? And we do have co sign if you d you. So that means that we can just substitute again. So we get the negative stays in front turns the integral of the Devi or, in our case, doing this anti derivative. This is just negative. The squared over two plus c. So now we have two sets of back substitution ZX two D&#x27;oh! So plugging in for V, we get negative. We said V was equal to side of you, but we&#x27;re still not done because original answer was in terms of data. So plugging in for theta, which we said was won over data plugging for you. Rather, this is equal to a negative sign of won over Seita. And that whole thing was squared, divided by two plus C. And this is our final answer.

Problem

Evaluate the following integrals. $$\int \frac{1-…

Problem 39 Hard Difficulty

Evaluate the following integrals.
$$\int \frac{d \theta}{1+\sin \theta}$$

Answer

tan ? - sec ? + C

Related Courses

Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Anna Marie V.

Kayleah T.

Caleb E.

Samuel H.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 8

Video Transcript

William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals

Anna Marie V.

Kayleah T.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

tan ? - sec ? + C

Calculus Early Transcendentals

Anna Marie V.

Kayleah T.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

William Briggs, Lyle Cochran, Bernard GilletCalculus Early Transcendentals

Anna Marie V.

Kayleah T.

Caleb E.

Samuel H.

William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals