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Current time:0:00Total duration:5:56

hopefully we all remember our good friend the chain rule from integral Safari from differential calculus that tells us if I were to take the derivative with respect to X of G of f of X G of let me write those parentheses a little bit closer G of f of X G of f of X that this is just going to be equal to the derivative of G with respect to f of X so we could write that as G prime of f of X G prime of f of X times the derivative of F with respect to x times F prime of X and if you want to see it in the end in the other notation I guess you could say it would be you could write this part right over here as the derivative of G with respect to F times the derivative of F the derivative of F with respect to X and then that's going to give you the derivative of G with respect to X this is just a review this is the chain rule that you remember from or hopefully remember from differential calculus it's hard to get it's hard to get too far in calculus without really grokking really understanding the chain rule so what I want to do here is well if this is true then can't we go the other way around if I wanted to take the integral of this if I wanted to take the integral of G prime of f of X G prime of f of X times F prime of X DX well this should just be equal to this should just be equal to G of f of X G of f of X and of course whenever I'm taking an indefinite integral G let me make sure the same color G of f of X so I just swapped so as I'm going the other way so if I'm taking the indefinite integral wouldn't it just be equal to this and of course I can't forget that I could have a constant here now that might have been introduced introduced because if I take the derivative the constant disappears and so this idea you could really just call the reverse chain rule verse reverse chain the reverse chain rule which is essentially or it's exactly what we did with you substitution we just did it a little bit more methodically with you substitution and we'll see that in a second before we see how u substitution relates to what I just wrote down here let's actually apply it and see where it's useful and this is really a way of doing u substitution without having to do u substitution or doing u substitution in your head or doing u substitution like problems a little bit faster so let me give you an example so let's say that we had and I'm going to color code it so it jumps out at you a little bit more let's say that we had sine of X and I'm going to write it this way I could write it so let's say sine of X sine of x squared and obviously the typical convention the typical the sine of x squared the typical convention would be to put the squared right over here but I'm going to write it like this I think you might be able to guess why sine of x squared times cosine of x times actually I'll do this in a let me just in a different color times cosine of X times cosine of X so I encourage you to pause this video and think about does it meet this pattern here and if so what is this indefinite integral going to be well let's think about it if f of X is sine of X what's the derivative of that what's f prime of X well F prime of X in that circumstance is going to be cosine of X and what is G well G is whatever you input into G squared so what's this going to be if we just do the reverse chain rule well this is going to be well we take oh sorry G prime is taking whatever this thing is squared so G is going to be the antiderivative of that so it's going to be taking something to the third power and dividing and then dividing it by three so let's do that so the if we essentially take the antiderivative here with respect to sine of X instead of with respect to X you're going to get you're going to get sine of X sine of X to the to the third power over three and then of course you have the you have the plus C and if you don't believe this just take the derivative of this you'll have to employ the chain rule and you will get exactly this and you say well what how does this relate to you substitution well a new substitution you would have set u equals sine of X then D you would have been cosine of X DX and actually let me just do that actually might clear things up a little bit you would set this to be U and then this all of this business right over here would then be D U and then you would have the integral you would have the integral u squared u squared I'd have to put parentheses around it u squared D u u squared D u well let me do that in that orange color u squared D u well that's pretty straightforward this is going to be equal to u this is going to be equal to u to the third power over 3 plus C which is equal to what well we just said U is equal to sine of X you reverse substitute and you're going to get exactly that right over here so when we talk about the reverse chain rule it's essentially just doing u substitution in our head so in the next few examples I will do exactly that