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Current time:0:00Total duration:6:16

Rewriting before integrating: challenge problem

Video transcript

so our goal in this video is to take the antiderivative of this fairly crazy-looking expression or another way of saying it is to find the indefinite integral of this crazy-looking expression and the key realization right over here is that you this expression is made up of a bunch of terms and the indefinite integral of the entire expression is going to be equal to the indefinite integral of each of the terms so this is going to be equal to we could look at this term right over here and just take the indefinite take the indefinite integral of that 7x to the third DX and then from that we can subtract the indefinite integral of this thing so we could say this is and then minus the indefinite integral of five times the square root of X DX and then we can look at this one right over here so then we could say plus the indefinite integral of 18 square root of x square roots of x over X to the third DX and then finally and I'm running out of colors here finally I need more colors in my thing we can take the antiderivative of this so plus the antiderivative of X to the negative 40th power DX so I've just rewritten this in color-coded things so let's take the antiderivative of each of these and you'll see that we will be able to do it using our whatever we want to call the inverse of the power rule or the anti power or whatever you might want to call it so let's look at the first one so we have we have what I'm going to do is I'm just going to find the antiderivative without the constant just add the constant at the end for the sake of this one just don't we make sure we get the most general antiderivative so here the exponent is a 3 so we can increase it by 1 so it's going to be X to the fourth it's let me do that same purple color or the pink color it's going to be X to the fourth or we're going to divide by X to the fourth so it's X to the fourth over 4 is the antiderivative of X to the third and you just had this scaling quantity the 7 out front so we can still just have the 7 out front so we get 7x to the fourth over 4 fair enough from that we're going to subtract the antiderivative of this now at first this might not be obvious that this you could use our inverse power rule or anti power rule here but then you just need to realize that five times the principal root of X is the same thing as so this is the same thing as five times X to the one-half power and so once again the exponent here is one-half we can increment it by 1 this is going to be X to the three halves and then divide by the incremented exponent so divided by three halves and of course we had this five out front so we still want to have the five out front now this next expression looks even wackier but once again we can simplify a little bit this is the same thing let me do it right over here this is the same thing as 18 times X to the one-half times X to the negative 3 power X to the third and the denominator is the same thing as X to the negative 3 we have the same base we can just add the exponents so this is going to be equal to 18 times X to the 2 and 1/2 power 2 and 1/2 power or another way of thinking about it this is the same thing as 18 times X 18 times X to the 5 halves power to the 5 halves power did I do that right yeah negative 3 oh sorry this is negative 2 and 1/2 let me make this very clear and this is going to be the negative 5 halves power X to the negative 3 is the same thing as X to the negative 6 halves negative 6 halves plus 1/2 is negative 5 halves so once again we just have to increment this exponent so negative 5 halves plus 1 is going to be negative 3 halves so you're going to have X X to the negative 3 halves and then you divide by what your exponent is when you increment it so divided by negative divided by negative 3 halves and then you had the 18 out front and we obviously are going to have to simplify this and then finally our exponent in this term let me not use that purple anymore our the exponent in this term right over here is negative 40 if we increment it we get X to the negative 39 power all of that over negative 39 and now we can add our constant now we can add our constant C and all we need to do is simplify all of this craziness so the first one is fairly simplified we can write it as 7/4 X to the fourth now this term right over here is essentially negative five divided by three halves so five over three halves is equal to five times two-thirds which is equal to ten over three so this thing right over here simplifies to negative ten over three X to the three halves and then we have all of this craziness now 18 divided by negative three halves 18 divided by negative three halves is equal to 18 times negative two-thirds which is equal to well we can we can simplify this a little bit this is the same thing as six this is the same thing as six times negative two which is equal to negative twelve so this expression right over here is negative 12 X to the negative three halves and then finally this one right over here we can just rewrite it as if we want we could well we can't write negative 130 ninths X to the negative thirty nine plus plus C and we're done we found the indefinite integral of all of this craziness and I encourage you to take the derivative of this and you can do it using really just the power rule to take the derivative of this and verify that it does indeed equal this expression that we took that we took the antiderivative of