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# 𝘶-substitution: logarithmic function

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.D (LO)
,
FUN‑6.D.1 (EK)

## Video transcript

we are faced with a fairly daunting looking indefinite integral of PI over X natural log of X DX now what can we do to address this is use substitution a possibility here well for u substitution we want to look for an expression and its derivative well what happens if we set u equal to the natural log of X now what would D U be equal to in that scenario D U is going to be the derivative of the natural log of X with respect to X which is just 1 over X DX this is an equivalent statement to saying the d u DX is equal to 1 over X so do we see a 1 over X DX anywhere in this original expression well it's kind of hiding it's not so obvious but this X in the denominator is essentially a 1 over X and then that's being multiplied by DX let me rewrite this original expression to make a little bit more sense so the first thing I'm going to do is I'm going to take the pie to do that in a different color since I've already used let me take the pie and just stick it out front so I'm going to take the pie out in front of the integral and so this becomes the integral of and let me write the 1 over natural log of X first 1 over the natural log of x times 1 over X DX now it becomes a little bit clearer these these are completely equivalent statements but this makes it clear that yes you substitution will work over here if we set our u equal to natural log of X then our D U is 1 over X DX 1 over X DX our D U is 1 over X DX let's rewrite this integral it's going to be equal to pi times the indefinite integral of 1 over u natural log of X is U we set that equal to natural log of x times D u times D u now this becomes pretty straightforward what is the antiderivative of all of this business and we've done very similar to things like this multiple times already this is going to be equal to pi times the natural log the natural log of the absolute value of u so that we can handle even negative values of U the natural log of the absolute value of U plus C just in case we had a constant factor out here plus C and we're almost done we just have to unsubstituted you is equal to natural log of X so we end up with this kind of neat looking expression the anti this entire indefinite integral we have simplified we have evaluated it and it is now equal to PI times the natural log the natural log of the absolute value of U but U is just the natural log of X the natural log of X and then we have we have this plus C right over here and we could have assumed that from the get-go this original expression was only defined for positive values of X because you have to take the natural log here and it wasn't an absolute value but now we so we can leave this as just a natural log of X but this also works for the situation's now because we're taking the absolute value of that where the natural log of X might have been a negative number for example it was a natural log of 0.5 or who knows whatever it might be but then we are all done we have simplified what seemed like a kind of daunting expression
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