Integral of ln(x) with Feynman's trick!

so we are going to do the integral from 0 to 1 of natural log X DX now once we've gone through our first few years of calculus this is a pretty easy integral to evaluate so we're gonna have a little fun with it and say instead of using u substitution instead of using integration by parts we're going to restrict ourselves and say we have to use fineman's technique in order to evaluate this integral now fineman's technique which is also known as the Leibniz rule says if we define some function I of T as an integral in this case from 0 to 1 of some function of T and X so let's say we have natural log of T times X with respect to X if we differentiate with respect to T what we get is the integral from 0 to 1 again but we take the partial derivative of whatever is on the inside with respect to T and this usually allows us to turn our complicated function like a natural log into something easier in this case the partial derivative with respect to T remember a partial derivative just means that we pretend anything that's not T is a constant so in this case we say X is a constant and we differentiate with respect to T well the derivative of natural log of TX is going to be 1 over T X and then by the chain rule the derivative of T X well X is a constant so when we differentiate this we actually get X on the top notice X is on the top and bottom so we can cancel those out and we just end up with 1 over t DX well in this case we're integrating with respect to X 1 over T is just a constant so we'll end up with 1 over T times X evaluated at 1 and 0 which will give us the final result of 1 over T so that means that our I prime of T is equal to 1 over T and now we can integrate both sides if we integrate I prime of T DT and then integrate 1 over T DT then we'll get the answer of I of T equals well the integral of 1 over T is just natural log absolute value of T plus C this plus C is going to give us a little trouble you cuz in order to use fineman's trick for integration we have to know at least one value of this integral because if we know one value of the integral we can plug it in for example if we knew that I of 1 was equal to negative 1 well then we could plug in 1 everywhere and then say I of 1 equals natural log absolute value of 1 plus C and then we could solve for C because we can plug in i of 1 equals negative 1 but in this case we have I of T equals natural log T plus C but we don't know any values for this integral because that's the whole thing we're trying to figure out normally when we want to use the natural log of TX we like to have a plus 1 on the end of the natural log because when we do this we can say I of 0 well the I of 0 will be the natural log of 0 plus 1 and the natural log of 1 is of course 0 so we get that as our answer and this gives us a known value we can plug in here we don't have a plus 1 so we have to think about some other strategy to figure out how we can turn this into an I of T that we can get information about initially you might think well if we just need a plus 1 inside of this integral to make it work maybe we do a substitution and just let X equal u plus 1 then DX equals D U and we can write this integral as the integral from in this case negative 1 to 0 plugging in the bounds of natural log u plus 1 D U and then we can do this one with fineman's trick the problem here is this integral actually becomes very very ugly when we try to use fineman's technique on it in this form so this is actually not going to help us it's just gonna make the integral more complicated we're going to need some other method we're actually going to need to think outside of the box so here's the method that I came up with normally when we do find mins trick we start with some function I of T the we want to integrate and then we differentiate it to get something that's easy to deal with but maybe we want to do that in Reverse we start with a function that's easy to deal with and then differentiate it to get something that we want so let's think about that if we want our i prime of T to be the function that we want well what function when you differentiate it gives you natural log of X remember we're differentiating with respect to T so X is like a constant we think about the derivative with respect to T of something equals something times natural log X if this is a partial derivative what derivative has a natural log in the answer and the solution is X to the T if we differentiate X to the T with respect to T we get X to the T times natural log X and this as long as we plug in T equals zero is the integral that we want so let's apply that strategy instead of starting with this natural log we do the other strategy we let I of T equal the integral from 0 to 1 of X to the T DX and this is an integral that we know how to do we can just use the power rule T is a constant with respect to X so if we apply the power rule this equals x to the T plus 1 over T plus 1 evaluated at 1 & 0 well 0 to the T plus 1 is just 0 and if we evaluate it at the upper bound we'll just get one to the T plus 1 which is just 1 divided by T plus 1 so I of T equals 1 over T plus 1 on the other hand if we differentiate I with respect to T if we do I prime of T what we get is the derivative with respect to T of X to the T well that's going to give us X to the T times natural log X so I prime of T is this integral our goal is to find I prime of 0 because i prime of zero is the integral of natural log X since X to the 0 is just 1 but we know I of T equals 1 over t plus 1 so what is I prime of T well I prime of T is just going to be the derivative of 1 over T plus 1 which we can easily find that's negative 1 over T plus 1 squared so if we want I prime of 0 we just plug in 0 i prime of 0 equals negative 1 over 0 plus 1 squared of course 0 plus 1 is 1 negative 1 over 1 squared is negative 1 that is the answer to our integral the integral from 0 to 1 of natural log X DX equals negative 1 so when we want to think about using things like fineman's technique and other advanced things sometimes we have to think outside of the box a little bit if we understand the structure of how the method is applied sometimes we can do it in unorthodox ways that weren't how we were taught in class but they give us the answers that we need just like this

Channel: Mu Prime Math

Views: 621,275

Keywords: math, calculus, integral, integration, feynman, trick, technique, leibniz, rule, differentiation, under, sign, challenge

Id: GW86SShcYbM

Length: 7min 52sec (472 seconds)

Published: Mon Sep 09 2019