Evaluate the integral5/2 fe dz5/2 f dz =(Use € as the arbitrary constant )Enteryouansi erin Ine ans er Oox...

Use Exercise 52 to find $ \displaystyle \int x^4 e^x dx $.

Is in developing extra for four into the egg DX. We will go for the tabular method or he we assumed the if a differentiating function to be extra support for an integral function. Is it correct? Okay, by continuously and subsequently differentiating and integrating this we find for rescue. All right. Thanks. 12 x squared. Also to the project 24 8 also to the four x 24. The parade zero. Correct. Now we go for assigned method. This multiplication will be plus this will be miners. This will be again. Plus this b minus and plus Okay, So the integral we can write from here is simply stripper for into the eggs minus four x cubed intuitively forehead first 12 x squared. Intuitive eggs minus 24 x into it to the four eggs. Last 24 to B to the Power X. Let's see. So if we take comin into the bar, X council perform minus four. Makes this cube past 12 X squared minus 24 x plus 24. This So this is the integral

We're continuing on solving with integral. But in this case our interval has a variable in it. So we're not necessarily going to have a numerical answer, we might get another equation out. So let's look at this one. I have two variable intervals here. I'm going from 8-5 a. And I'm looking at one over X. Dx. So looking at the anti derivative, I have to figure out what give us one over thinking back to that, that would be my derivative of log X. That would give me that answer. So I now know my anti derivative and I can throw it into my equation now. So I would have The natural log of five a minus the natural log of A. I could leave it like that. But knowing log laws we can clean it up a little bit more. We know that natural log when I subtract them, they can be put together using division. So this guy can be put together, it would become five a. Over A. And then from there R. A. S cancel out. So this one, even though we have a variable interval because of how it's set up, no matter what I pick for a my answer will always work out the natural log five

We're continuing on solving with inter girls, but in this case are Interval has a variable in it. So we're not necessarily gonna have a numerical answer. We might get another question out. So let's look at this when I have to variable intervals here. I'm going from eight of five a. And I'm looking at one over X t. X. So, looking at the anti derivatives, I have to figure out what give us one over thinking back to that. That would be my derivative of Log X. That would give me that answer. So I now know my anti derivative and I can throw it into my equation now, so I would have the natural log of five a minus the natural log of a I could leave it like that. But knowing log laws, you can clean it up a little bit more. We know that natural log when I subtract, um, they can be put together using division so this guy can be put together. It would become five a over a and then from there are a cz cancel out. So this one, even though we have a variable interval because of how it's set up no matter what I pick for a my answer, will always work out the natural log Fine.

Reference. We have to evaluate that given definite integral that is given integration of 2.7 to 5.3. Lower limit is 2.7 and the upper limit is 5.3 and the expression is one upon extra decks over and upon X. Into road X plus four. Yeah next. So this will be costume 2.7- 5.3. Mhm but upon extra deck so we can add it back to the power of monastery wide Plus four of BX. So integration of this will be a cost to. Actually the parliament is three Y two plus one Upon Monastery Y two Plus 1. Let's four X. Yeah 2.72 2.7- 5.3. So this will be a costume X. To the power minus vanuatu minus one way to upon money is off to into two plus four X and are limited 2.7- 5.3. Yeah so we can put the limit so this will be caused to 14- 5.3. money is up to and to 5.3 to the power management were to Mhm one is a point fight minus off four and 2 2.7- of 22.7 to the power of -15. Yeah, So first so from the calculator, we can solve it. So this will be a cost to 10.7484. So this is the answer. I hope you understood. Thank you.