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Integrate mark proof

hi it's time for another math easy solution turn discuss integration by parts and over a proof of this method basically every differentiation rule has it has a corresponding integration rule for instance the substitution rule for integration corresponds to the chain rule for differentiation and you see my video on substitution rule where I actually proved the substitution rule for integration using the chain rule for differentiation and now the rule that corresponds to the product rule for differentiation is called the integration by parts and now the product rule states that if the functions F and G are differentiable meaning you could take a derivative of them then if you have the derivative of a product of the two for example D over DX of f of x times by G of X this would equal to by the product rule this is going to equal to f of X G prime of X plus now we have a plus this is going to be F prime of x times G of X so as you can see this one is just going to be addition of the two products but in in one case you'll have a derivative of G of X the other case derivative of f of X yes so now if we were to integrate both sides so that we write them as indefinite integrals will have something like this D over DX of f of X G of X and DX always at the end of an integral equals two now we're integral on the right side f of X G prime of X DX plus integral here of F prime of X G of X DX and now when you have this on the left side here is since integral is just an antiderivative so antiderivative of a derivative it's just going to be the inside function so this would equal to f of X G of x equals 2 now the integral on this side I will just write down the same side right there G prime of X DX plus F prime of X G of X DX now in this case right here like when you take an integral you always have to add a constant but in this case since we already have more integrals and when you take those there's gonna be more constants and when you add them up you're gonna have a constant anyway so we're just gonna ignore that constant what you have to do a plus C until later whenever you evaluate the integral then you have to add them and when you evaluate all the integrals and you have to add the C so now what we could do here is basically develop now the integration rule by parts so we would look at this function right here and write everything in terms of this one here so we'll move this to the left side so we will have integral of f...