Integration by parts: ∫ln(x)dx

the goal of this video is to try to figure out the antiderivative of the natural log of X and it's not completely obvious how to approach this at first even if I were to tell you to use integration by parts you'll say well a the integration by parts you're looking for the antiderivative of something that can be expressed as the product of two functions it looks like I only have one function right over here the natural log of X but it might become a little bit more obvious if I were to rewrite this as the integral of the natural log of x times 1 DX now you do have the product of two functions one is a function a function of X it's not actually dependent on X it's always going to be one but you could have f of X is equal to 1 and now it might become a little bit more obvious to use integration by parts integration by parts tells us that if we have an integral that can be viewed as the product of one function and the derivative of another function and the derivative of another function and this is really just the reverse product rule we've shown that multiple times already this is going to be equal to the product of both functions f of X times G of X times G of X minus minus the antiderivative of instead of having F and G Prime you're going to have F Prime and G so f prime of X F prime of x times G of X G of X DX DX and we've seen this multiple times so when you figure out what what should be f and what should be G for f you want to figure out something that it's easy to take the derivative of and it simplifies things possibly if you're taking the derivative of it and for G prime of X you want to find something where it's easy to take the antiderivative of it so a good candidate for f of X is natural log of X if you were taking the derivative of it it's 1 over X let me write this down so let's say that f of X is equal to the natural log of X of X then F prime of X is equal to 1 over X and let's set G prime of X is equal to 1 so G prime of X is equal to 1 that means that G of X could be equal to could be equal to X and so let's go back right over here so this is going to be equal to this is going to be equal to f of X times G of X well f of X times G of X is X natural log of X so G of X is X and f of X is the natural log of X I just like writing the X in front of the natural log of X to avoid ambiguity so this is X natural log of X minus the antiderivative of F prime of X which is 1 over x times G of X which is X which is X DX DX well what's this going to be equal to well what we have inside the integrand this is just 1 over x times X which is just equal to 1 so this simplifies quite nicely this is going to end up equaling this is going to end up equaling let me I can go let me put it right there this is going to end up equaling all right X natural log of X natural log of X minus the antiderivative of just DX or the antiderivative of 1 DX or the integral of 1 DX I should say the antiderivative 1 is just minus X and this is just an antiderivative of this if we want to write the entire class of anti derivatives we just have to add we just have to add a plus C here and we are done we figured out the antiderivative of the natural log of X I encourage you to take the derivative of this for this part you're going to use the product rule and verify that you do indeed get natural log of X when you take the derivative of this