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Find the following integrals.
$$\int(x+1) \sqrt{3 x+2} d x$$
In discussion. We need to find the value of integration. X plus two Upon underwrote after Squire Plus four. X plus 13 D. X. Let's see how the whole discussion consider X Plus two equals 2. Three attention to you. Therefore by the documentation we can right do you have physicals too three? He can't inquire you do you now substitute the value of expressed to and dX in the above integration. So the integration becomes integration. Three tangent. You Yeah into three. She can't esquire you day you upon underwrote And we can write access problems for XPS. 13 has X-plus 2 to the power to plus nine. Now substitute again Expressed with close to three tangent. You so the integration becomes integration three tangent you into td She can't esquire you do you upon underwrote three Squire and gently Squire you plus nine. So the silver calls to integration nine tangent to you so you can't esquire you d you upon three under route Tangent Esquire You Plus one. We know that one plus tangent squared meters. Also frequent squared heater hands integration becomes integration. Nine tangent. You he can't esquire you do you upon three second. You on for the solving we get integration three attention to you. He can't you do you since the integration of seek and you tangent. You do you with calls to c can do you hands. The above integration becomes three second. You since in the beginning of questions we have already considered X-plus two is close to three tangent. You hence the value of tangent you will be called to X-plus two upon 3 no square the both sides. And Erdogan for the Sylvie calls to one plus tangent Squire you it recalls to nine plus access Choir plus four x plus four upon nine one plus tangent squared it has cause to see Kansas Choir data hands the value of second you will be called to one upon 3 under route X squared plus four X plus 13 hands the value of integration X-plus two upon underwrote access Choir Plus four x plus 13 dx will be close to underwrote access Choir plus four x plus 13 plus a constant of integration. So this is a final answer for this problem. I hope you understood resolution. Thank you.
In the problem will have to find it. You moved off DWells plus one days for solving these will assume twits plus one as t. So it become twice the X people that DD or Haiti's Indication T minus 1/2 in due to the power 1/3. Did he want to? So become one upon four. Indication inter t to the power upon me. Dele minus one upon four. Integration. Keep the power on upon three d. D. Now here has become one Obama for indication. Give the power for on three D D minus one upon for integration. Two to the power one upon treaty. So this become 3/4. Give the power 7/3 bonds seven minus three upon four Q to the power for a boundary over full plus c no. This becoming 3/28 in the two. It's plus one to the power off. Three upon. Sorry, this is a seven country minus 3/16 you twigs plus one to the power for country. Let's see. So this is the answer
Let's evaluate the following integral and let's go ahead and take the fraction inside the inte grant and go ahead and rewrite this using partial fraction decomposition. This is what the textbook would call case one. We have distinct linear factors in the denominator, so we can rewrite. This is a over two X plus one plus B over X minus one and then let's go ahead and multiply both sides by the denominator two X plus one x minus one. So we multiply on the left, the two X plus water and the X minus ones will cancel. So we're left with five x plus one on the right, where we multiply by this fraction by this product of linear polynomial. The two x plus ones will cancel when we multiply A. So we have a X minus one and then similarly for B, the X minus ones will cancel. And so we left over with two X plus one. So let's go ahead and rewrite the right hand side. So let's factor out an X here, and then we have a and then to be, and then we have B minus a. So if these two expressions are equal, then that must mean that the terms in front of the X term are equal and also that the constant terms are equal. So this gives us two equations. We have a plus to be equals five. But then we also have that B minus A is one. So we have a two by two system for A and B. Let's go ahead and solve that many ways To solve this one way is to just take this equation over here in green, solid for B and then go ahead and plug this be value into the other equation. So if we do that a plus to be replace, be with one plus a this is equal to five. But then we can go ahead and simplify this. We have three a equals five minus two. So that means a is one. And then using this equation over here, we also get that B as to so I'm running out of room here, let me go into the next page. So now the next thing, actually, before that what we should do here is go ahead and replace a and B with the values that we found. So now We'll replace A and B with one and two and then we'll integrate. Instead of integrating the left hand side, we'll go ahead and integrate the right hand side. So we have a was equal to one and then be was equal to two and now we have two. Integral to deal with. It may help you here to use a U sub. If you'd like to use the use up here, you can let you be two x plus one for this integral here. If you don't like that minus one on the X, you can go ahead and do another sub here. Let's do w this time then D w equals d X do u equals two d X so you can go ahead and carry out the U substitution here to evaluate this Integral. In either case, we are going to see natural algorithm. So for the first in overall, we get the one half natural log Absolute value two x plus one and for the second, integral to natural log Absolute value, X minus one plus our constancy. And that's our answer
Indeed your question you have to integrate the function and the companies three X -2 Divide wife. Hello. To express. What? Okay. In the X. So I substitute here to Express one by days where Then I get to d. x. equals two 2 82 Counselors. And then they exposed to DDT anything. No as a substitute the substitution. So it's a pleasure to express. We're here to put these world Under the place of the X. We have 250 me guilty and it'll be place of X. I can say that I have to put here from the Square -112 The Explorer -1 Voter and minus two. No this TNT cancel out and after I think we get here 380s were -3 By 2 -2. Into do you think? No I didn't say that if I take calcium too then I got here 30 teeth would -3 -4 DT. I'll take this to outside Then I get three days were -7 in duty. No, I integrate. Then I get three d 2 x three minus 70. Let's see. No three Year Counselors. And after that I, but the value of T premier, the velocities wrote off to explain as well. Okay. Say it provides me to express money to the power three Baidu minus 17 to add sweetness going to the bar when my two. Let's see. Okay, this is my final answer here. Thanks.