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Intejate # Clloig by Substituhbn: sh?& Cs xdx...

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Intejate # Clloig by Substituhbn: sh?& Cs xdx

Intejate # Clloig by Substituhbn: sh?& Cs xdx



Calculus 1 / AB
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Using integration by parts. $$\int e^{-y} \cos y d y$$

Next over us all the question # 26. And in this question we need to evaluate the integral which is integration U. To the Power X. Contingent into the power X. Dx. Now to solve this problem we're going to use the substitution method and we're going to substitute. It is about X. S. U. On differentiating both the side will get a to the power X. Dx. Is it cold studio? So this integration would become no integration tangent. You do the power X. We have taken as you and ate the politics doctor Dx. We have do you. So we have got this Now we can easily evaluate this integration. We know that the integration of changing to us natural lock seeking to you plus see where the sea is a concern of integration. Now can I get put the value of you here so we'll get natural lock 2nd. You we have it the politics plus C. So this required to answer for the given integration.

In this fusion that given integral is Yeah that is integral to the power eggs six squared off into deep or x. D. X. So this integral is given. And knowing the community saying that we have to follow this Cuban integral. So let's start asking this question. So here I will apply a method that is called substitution method. She by this method I will assume that okay Each of the four x is equal to you. No take a differentiation on both sides. So it will be written as it is. Yeah. Each of the four X. DX is equal to do you. So this is a step number one and this is a step number two. So by using this step number one and number two, the given integral will be written as that is integral to the port X 2nd. 2nd score off into the ball X. Mhm Dx is equal to integral here. This report X D X will be written as that is by using a step number two, I can say that do you. And sag six squared off into the products were written by using the system number one. I can say that second square do you. So no longer result. It means that given integral is changed in you variable so no, my next step is I had to get the answer for integral sex square. You D You So here I will use a standard integration formula. That is formula is integral 60 square You do you is equal to 10 you plus c. So use this extended formula. So I can say that in a bush tape so I can see that the result will be euthanized studies integral eat to depart X stratosphere off into the part. X. Dx is equal to Danny. You plus see now next step is we have to get the answer in original X variable. So it means put the value of you by using a step number one so I can see that value is, that issue is equal to it to the power X. So after replacing this You by each of the 4x. The result will return as studies. Given integral is integral to the to the power ex. Except square off to the power X. The X is equal to again it to the part of, sorry, 10 of it to depart X plus C. So this is the asshole for a given condition. Thank you.

Now in this question we have to evaluate the given and rule which is integration. You do the politics hyperbolic Hussein 2 -8. The politics bx. Now to solve this problem we're going to use the substitution method and we're going to substitute 2 -8 of the politics as you no differentiating on both. The sides will get minus into the power X. Dx will be cool to do or we can write that into the power X. Dx is equal to minus do you. So this integration would become integration. The hyperbolic Cosine 2- into the politics. Now this thing we have taken us you so it will be hyperbolic cosine you and a to the power dx we have minus to you so that when we minus to you now we can easily evaluate this integration. So we'll get minus hyperbolic sine. You plus C. Where C is a concert of integration. Now we're gonna replace the value of you here so we'll get minus hyperbolic sine. You we have to minus into the politics plus C as it is. So that's a great answer for the given integration.

With this integral. We're going to be using integration by parts to reduce this P. Two. The fourth term down to just one. And so we're going to be doing this um four times here. So use equal P. To the fourth to use equal to four P. To the third. And we can see that we're already reducing the power here as we're going to have, this is equal to U. Times V. So it would be negative peter the fourth times E. To the negative P. And then minus the integral of E. Times D. U. And now we can see that we have peter the third here. So you've effectively just reduced this power down by one. And we're gonna have to keep doing that until we get um The power down to zero. So so again we're doing integration by parts here. You is going to equal well first I'm actually going to take out this um negative four here. So we're gonna have plus four times the integral peter. The third time's eat some negative P. And so now we can let you hear equal peter. The third do you? Is equal to three P squared T. V. Is equal to eat the negative P. So then V. Is equal to negative E. To the negative P. And go ahead and move these over here. And so we still have this negative piece of the fourth of the negative P. Term. Now we have U. Times V. So negative peter the third. Eat the negative P minus the integral of either the negative P times three P squared. So I'm actually going to just take this negative one out. So we're gonna have plus the integral of three P squared times E. To the negative P. And I'm actually just going to take this three out of this integral here as well. So I don't have to to another step there. And so now we're going to do the same thing here. Let U equal P squared, Do you is equal to two p. D. V. Is equal to eat the negative X. So then V. Is equal to negative E. To the negative X. Just going to go ahead and put these over here. And so this is equal to negative P. Of fourth. Either negative P plus four times negative P. To the third. Yeah the negative P plus three times negative peace squared here. The negative P. And then this is gonna be plus the integral and I'm actually gonna take this to out here. So plus two times the integral of P times eat the negative X. Dx. And so you can see this kind of pattern forming. Um And there is a general formula for reducing um terms like this. However I'm not going to go into it here. Um And so this last integration by parts we're gonna let u equal P. Do you equal dp tv equally to the negative X. So then V. Is equal to negative E. It's negative X. And go ahead and say that this is equal to negative P. The fourth. Eat the negative P plus four times negative P. To the third. Each the negative P plus three times negative peace squared each the negative P. And then plus two times negative P eats the negative pd. Should have been piece here. I'm not excess. That's okay. Um And then we have minus the integral which is actually just gonna plus the integral of E. To the negative X. And so this is equal to negative p. To the fourth each negative p plus four times negative P. To the third E. To the negative p plus three times negative P squared each the negative P plus two times negative P. Each the negative P. And then this is going to be minus again. I should have put a P here and our necks and this is gonna be minus E. To the negative P. And we can go ahead and just put the plus C in here. And we have 123 parentheses and 123 in parentheses. And so this is our integral.

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