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What techniques of integration would you use in order to evaluate the integralYou should just list these techniques, you dont have to evaluate the integral....

Question

What techniques of integration would you use in order to evaluate the integralYou should just list these techniques, you dont have to evaluate the integral.

What techniques of integration would you use in order to evaluate the integral You should just list these techniques, you dont have to evaluate the integral.



Answers

Give some examples of analytical methods for evaluating integrals.

Yeah, So I asked a list off. What are analytical methods we use for integration? Okay, well, one cases where? Sure about this. Where you have known anti derivatives PLO, for example. You know that when you integrate the sine function, you get minus the co sign functions because, you know, the derivative of coastline. Okay, so things that you know when you know the derivatives therefore, you know, the anti derivative. That's one thing. Ah, the other thing that we do are we make, you know, substitution. So, for example, if you were looking at you and integral that had, um you know, would you say, um, you know e to the X squared times X, You know, DX, you know, when you can make substitution is for sore About that, that's a bad example. Um, that's gonna be integration by parts. But when I see something, let me just show, For example, unless you say if you see something like you two x plus one times x dx. What you see here is a function, and I can look and say a derivative so something like if I had and I'm doing a bad example, I might have to redo all of this ship. We're asked the list off different methods we've used for analytical evaluation of intervals. So just ah, off the top of my head. One is where you have known derivative. For example, you know that when you take the derivative off the coastline function, you get minus the sine function. So that tells you that any time you need to integrate the sine function, you're going to get minus the co sign. So if I can recognize Oh, I know the derivative that function, Um, then I know anti derivatives. Other examples of things we can do our with substitution. So, for example, when you see something like X over X square plus one d X, any time you can organize an integrated into a function and it's derivative, then that makes it easy to work with. So, for example, of you is equal to X squared plus one, Then do you is equal to two x dx so I can write a to here and a 1/2 here and then this just becomes the integral of 1/2 one over you, Do you, which is much easier to integrate, So substitution are another thing that we do, um, beyond that other things we do are we have dealt with what? Integration by parts. And if you're ever alone on a desert island, you don't remember this one. It's just unwrapping the product rule. So you know that when you take two functions, you envy their derivative UV plus the inter go of uh, v d u um or so I read that Wrong, um, gonna be you prime B plus the prime you. So that tells me that I can integrate both sides of this and that just gives me that If I wanted to solve for right here, that would tell me the integral of you. The prime is equal to U V minus the integral of V d u. So that gives us integration by parts. Other methods that we've done, um, have included, say, partial fractions. So that's when you're dealing with some rational expression s o. When you see something like, you know, x squared plus two over X plus one DX. Go ahead and performing this, um, division said that you can get this into separate intervals that are easier to integrate. Um, other methods we've done say completing, completing the square. So you see the integral say one over X squared plus two x plus three d x. You're rewriting this with completing the square and then recognize and perhaps a trick substitution or unknown Integral. So these are all just examples of analytical methods. We've been so far to deal with integration.

Integration off one plus 10 x and say kiss critics X Select one press 10 x equal difference 30 Respect Wicks. So sick squared X DX It will do. Do you think? Very vigorous will convert as okay. So it is his great Berto less but the value up PS plus 10 X described by

Invigoration off one plus Dennis Correx by annex As we know that one plus 10 square X is equal to shake a square x bx take his critics. So finally it is Take a square x the X by panics No, Let an X equals to t and Defense city Respect to X Second score Extra years will be so Finally it is DT Bharti So loved tea Plessy put the value of e J Dan Nix Love Dan Nix Let's see.

Seven nut to problem for They're asking us to explain, How do we use integration by parts to evaluate a definite integral. So our integration by parts formerly a tells me that if I have something that's in the form of you of X and then Devi, that's going to be you of acts vfx minus the integral vfx you prime of x dx or Iran about rent out a room there. So let me just make that little bit clearer. So minus the integral of VT years of the FX you prime of x dx. So the only thing that changes with limits of integration issues you have to recognize that this expression just becomes you of eggs vivax evaluated from a to B minus the definite integral of the the use of the FX you prime of x dx. So that's how you would handle the definite integration were using integration by parts


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