an-integrand-with-trigonometric-functions-in-the-numerator-and-denominator-can-often-be-converted--4

Question

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution $u=	an (x / 2)$ or, equivalently, $x=2 	an ^{-1} u .$ The following relations are used in making this change of variables.
$$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$
$$	ext { Evaluate } \int \frac{d x}{1+\sin x+\cos x}$$.

Tom Greenwood · Accepted Answer

Okay, We&#x27;re gonna integrate this using the formulas they gave us at the top of the section of problems. And so we can replace the D X by a two over one plus u squared, do you One state of one, The sign of X gets replaced by to you over the one plus u squared and the co sign gets replaced by one minus. You square over one plus u squared. Now I&#x27;m gonna multiply the top and bottom of this by the common denominator one plus u squared to get rid of all those complex fractions. So my numerator is just gonna become too do you, denominator? One times the one plus u squared the two you fraction divides out I just get close to you and my other fraction that divides out again I&#x27;m left with the one minus you squared So now I combine my like terms down on the denominator And I end up with to do you over two plus two You and I couldn&#x27;t divide all those twos out And I just get the end girl of do you over one plus you And hopefully you recognize that we can have our natural algorithm formula here for this. This is you, TV over V or you know, the one over the TV type of formula. We&#x27;re betting the function be the one plus you. So this will integrate and me the natural order of them. Uh, the absolute value have one plus you Plus are arbitrary integration constant of C and now we substitute in the U value. From the beginning of all this, that&#x27;s the tangent off the X over two. So are finally answers The natural log Absolute value one plus tangent of X divided by two waas see

Tom Greenwood · Answer

Oh, okay. We&#x27;re going to integrate this using the substitution ins that we have from problem number 87 were given it. Stop Toughness Group of problems. So DX is to over one plus u squared times, do you and two plus co sign of X becomes two plus one minus You squared over one plus u squared. So to get rid of the complex fraction I&#x27;m gonna multiply the top and bottom by the denominator of one plus u squared Oops, you squared I&#x27;ll get rid of my complex reactions So the numerator just the one place you squares divide out and I&#x27;m left with to over to you in the denominator I&#x27;ve got two times the one plus u squared So two plus two you square plus and then they divide out I&#x27;m left with the numerator one minus you square. So combining the like terms and pulling out my constant here of the two and numerator I have to times the and a girl do you over you squared plus three And if you recognize that form that&#x27;s in the form of one over you squared plus a squared which has it roll off inverse tangent so this will be to times. Now, This a is important in our integral The formula for the girls Actually one over a times the inverse tangent of you over a. So if a squared is three, that means a s the square root of three. So I have one over the square root of three times the inverse tangent of my you over a again the square root of three. We&#x27;ll have our arbitrary constant of policy that will add on there in a second. And now I can substitute Because of the start of all this, They told us that you equals the tangent of X over two. So we have to over a square with a three times the tangent. It&#x27;s our inverse tangent off a tangent of X, divided by two all divided by square with three and then our plus

Tom Greenwood · Answer

okay. Going to solve this using our different substitution is that we were given at the start of this group problems the D X we can substitute that as to over one plus u squared times to you and the denominator. We have one minus the co sign which is one minus. You squared over one plus u squared I&#x27;m gonna multiply top and bottom by the common denominator of one plus u squared to get rid of my complex fractions. So that just gives me the integral of to do you in the numerator and in the denominator one times the one plus u squared and then minus the one minus you squared. And so now we just combine like terms and we end up with the integral of to to you over to you squared and the 20 divide out. So that&#x27;s the integral of do you over you squared or you to the negative second, do you? And we can integrate that just with the power rule. Adding one to our exponents and dividing by the new exponents gives us negative you to the negative first power and now making the substitution back in where you equal to Tangent of X over two. We have negative tangent of X over two to the negative first power. Then we can make that negative one over tangent of X over two or if you like, Here&#x27;s a reciprocal trick ratio, and that would be the negative co tangent, uh, hex over to. And don&#x27;t forget your plus C whatever form you&#x27;re going to eat.

Esraa Samir · Answer

using exercise 87 Integration Oh, the eggs over one plus sign X plus Resign X equal X equal toe 10 for negative one You the X equal toe over one. Pursue square do you and resign x equally one minus. You&#x27;re square over one plus u squared When sign X equal to you over one us you square Equalling integration to overrun plus u squared do you over one plus to you over one plus you square plus one minus you squared over one plus u squared equal integration to over one plus you square over one plus u squared, plus to you plus one minus You square over one plus u squared Do you sequin continuation toe over one plus you square or good to for us to you over one plus u squared Do you equal integration? 1/1 Plus you Do you equal Lynn one. Bless you Close Seen Make substituting X equal to 10 for negative one. You and you equal then x over two. So Lynn and one plus 10 eggs over to Yes, see

Esraa Samir · Answer

from racial function. Off degree metric functions we have substitution takes equal to 10 four minus one You the x equal to do you over one plus u squared sign X equal to you over one plus you square oh, equal integration Oh, so do you Over one plus you square over one close to you over one plus you square Quinn, Indication toe do you over one plus you square over one plus you square does to you over one plus u squared Couldn&#x27;t integration to over one plus u squared Do you substitute one issue sequin you Do you, sir TV and do you equal leaving equal to litigation one over the square dd quit to integration People were negative Toe Devi instigation off exporters e the X equal export a plus one over a plus one Let me we have tool boy Be bored Negative too Plus one over minus to plus one then substitute Thank you Equal Be equal one plus you Then we have minus tool over one bus You will see substitute back You equal 10 x over two thing It&#x27;s equal minus 2/1 Close 10 x over two Let&#x27;s see. This is finally

Esraa Samir · Answer

using exercise number 87 Litigation. Well, the X equal over one minus cuisine. Ex frequent X equal to 10. Minus border minus one. You the X equal to over one pursues. Quit. Do you cosigned? X equal one minus you. Square over one Might one plus U squared equal integration toe over one pursues square, Do you over one minus one minus. Use queer over one. Plus you square Equal instigation to over one assumes group over one plus u squared minus one minus you square or were one plus you screwed, do you? He quit instigation toe over one plus loose queer over toe You squared over one plus u squared Do you equal integration one over you squared you equal minus one. Were you close? See bake substituting X quick eggs equal to then bowler Get to warn you and your equal saying x over toe this equal minus 1/10 x over two plus seeing equal minus dying x over two plus c

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An integrand with trigonometric functions in the …

Problem 90 Hard Difficulty

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