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Problem 24 Easy Difficulty

Substitutions in single integrals How can substitutions in single definite integrals be viewed as transformations of regions? What is the Jacobian in such a case? Illustrate with an example.

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{'transcript': "So how did he use the integration by parts to integrate a single term That does not follow the basic immigration rules. I mean, how do you do that? I mean, it can't It can't be done. So, for example, have a single term like the natural love of X d X. I mean, this is not follow the basic immigration rules. Just a single turn. You can use the integration by parts to actually find this one here. Okay, so basically, this is gonna be I mean, the integration by parts formula is you. Devi, if you're integrating something like that, that is gonna be you ve my integral. You just stated this one just verified approved it in a previous video. So you have any function you and then the derivative of another function V integrate them. You're gonna get You can use this one to integrate them. Okay, so here I could take this natural log effects. As may you and think the derivative. Hear us my TV. Okay, then I can do this for so once you were able to do that, then this is gonna give you, uh, u v rus natural. Ah, relax and my V here. Okay, Is gonna be X? Okay. You have the eggs and you innovate. It is he's gonna be X. And then my eyes integral of the V s X and this Do you? I need to find a derivative of this one. And that is just one of her ex. Yes. Now, those one ISS can be, uh, simply integrated because this is gonna be ex national out of banks. Now, this is gonna cancel this one. It's just gonna be integral off one p x. And that is just x. Not a lot of ex miners. This is just x us a arbitrary constancy. See? So this is the integral for this right here. So with innovation by parts were able integrate this this single turn right here that does not follow the immigration rules because there is no way we can we can find an integral of this one. But the integration by parts were able to thio integrate this one and found out that this is actually the answer to the question. So that is how you can use innovation. My parts you evaluate in a single function, we'll turn. That does not follow. Uh uh, the immigration rules"}

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