5

Ccde ae64x (1+n)"axTz sunlx Cuj X dx (,...

Question

Ccde ae64x (1+n)"axTz sunlx Cuj X dx (,

Ccde ae 6 4x (1+n)"ax Tz sunlx Cuj X dx (,



Answers

$$\int \frac{-6}{x(4 x+6)^{2}} d x$$

In this question we have to integrate the function that is signed ways to the Power four of 6 X. First of all we can write it as sinus square six X. Rest of the power to. Okay now we apply the formula which says that Sinus square the attack and the button is one minus cost twice of theater, divide by two. So after applying this formula we can see we get here one minus cause of twice of this angle means 12 wax And divide by two and Holy Square. Okay now I'm going to open here Holy Square so I should get here in the numerator one plus courses square well wax and minus two costs well works and divided by four year. Okay now we can see that there is cause squared um and it should be again replaced here by the formula that is one plus cost to theater by two. So make these changes here one x 4 I'm going to take it outside then I remained with one plus discourses where time is replaced by one plus cost 24 X divided by two and minus two. Cause well works is there? Okay, so these are the things which have here now we can see that I am going to take Here product of one x 4. Okay then I what I get here, you can see here and that is one x 4 and right, I'm going to write its integration and plus we have to multiply by one x 4 again. So we get here integration of one x 8 and then we have integration of one x 8 Course 24 x and minus. Then we get here the integration of one x 2 cost 12 works. Okay, now we can into their dates of them. Integration of one by four is one by four X. Integration of one by eighties, one by a tax integration of discourse. 24 axes signed 24 X, divided by 24. And integration of course 12 axes signed 12 ax, divide by 12 plus C. Okay, now we can see that we have to simplify after integration. We can add x by four And X x eight after adding these similar terms I should get here three x by eight and we can see that this term turns out into mhm Plus off one by +91 92 signed 24 X and minus of one by 24 signed 12 X plus C. Okay, this is the answer for the given question here. Thank you.

The questions is that we have to find the integral for cause of four X into cause of six x. Dx now moving towards the solution. So for the governing trigger as we know that too into because of A into cause of B as opposed to because of A plus B plus because of a minus B. Therefore cause of four X into cause of six X can be written as one by two of cause of connect Plus cause of two x. So the integral will be integration Of one x 2 into cause of 10 x less cause of two weeks. So this can be written as I suppose to win by two s common integration of Because of 10 x bless Cause of two weeks dx. This will be called to one x 2 and to sign of Xanax by 10 Less sign of two weeks by two. Let's see. And this will be the solution to the question. Thank you.

Let's use partial fraction to composition to evaluate the definite integral from zero one. Looking at this fraction, we see that there's a quadratic in the denominator. So let's go ahead and try to factor that. So does this denominator factor. Sure does weaken. Take X minus two X minus three. So this is what the book would call case one. We have a distinct linear factors non repeated. So we have hey, over X minus two plus B over X minus tree. So this is the form for the partial fraction to composition, and I will have to use some algebra here to sell for Andy. So looking at this equation here circled in red, that's good and multiply both sides by the denominator on the list. So multiplying by X minus two times X minus three. On the left, that denominators will will cancel. And on the right hand side, we have a minus three plus B X minus two. Let's just go ahead and rewrite the right hand side. Let's pull out the ex term. We have X and then we have a Plan B and then we have minus three, eh? Minus two b. It's on the left hand side. We see that the coefficient in front of the ex is the one on the right. It's a plus, B. So we must have a plus. B equals one. Similarly, the constant term on the right is on the left. Excuse me is minus four, but on the right hand side, the constant storm is minus three minus two B. So this gives us a two by two system of equations too soft for A and B in that last equation there, you could go out and multiply by a negative one many ways to solve these systems. For example, we could take the first equation, solve it for Bea, and then we could plug this value of be into the other equation and in software, eh? And doing so will have three a plus two, one minus a equals four. And this will give us a plus two equals four so equals two and then must take this value of a plug it into this equation up here and we get B equals one minus two, which is minus one. So we found an B. So now we're ready to integrate. Let's come to the original problem the original integral. In the top left, we can rewrite this. So we replaced the fraction with the partial fraction to composition over here in the far right upper right corner. And we've just found A and B. So I have two over X minus two and then for B, we have a minus one over X minus three, and this new integral should be easier than the original. Now, if this X minus two or explain histories bothering you, you can go ahead and do a U substitution. U equals X minus two and for the second one, x minus three. And by doing so after simplifying, we should have to natural log absolute value X minus two minus natural log absolute value X minus three from zero one. Let me separate this from our scratch work. We've evaluated the inner girl. We have the natural logs. Now we just plug in the end points and simplify. So we plug in one first. We have natural log of negative one, but we have absolute value, so that just becomes a one. Then we have natural log of one minus three is negative to take the absolute value you just get into there and then when we plug in zero, the first term will become two times natural log. We have a negative to their absolute value. Makes it positive. And then finally, we'LL have a natural aga positive three. After we take absolute value and then just go ahead and simplify as much as you can. We know natural Aga one zero Combining these national log of twos. We'LL have negative three Ellen, too, and then watch out for this double minus here. That becomes a plus at one three. And there's many ways to simplify this. For example, we can write. This is Ellen three minus ln eight. By taking that using one of the log properties that says you can take the coefficient in front of the log, bring it inside. But then you have to raise it to the exponents. So two to the three is eight, and then I could use another property of logs log. If you subtract the lines, you can rewrite it as a fraction log of a fraction. So it looks like I have any one of these will be the right answer. Let's just maybe narrow down some of these two right here, and that's your final answer

What question? About 33 we gotta find the value of this indefinite integral. So for that first, we will use the constant rule off into differentiation where in the constant comes outside on the rest of the terms remain as it does now. This is an exponential function. And the entity we'll use the exponential rule off NT differentiation in which the integration will be here is to the poor attacks over eight, plus the constant of integration. Creationists see 6/8 is nothing but the common factors. Tools for two times three is 62 times for state. It is to the poor, it extremists as voters. Plus, we have a constant off integration as well. So this is the final answer.


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