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# Definite integral of rational function

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.B (LO)
,
FUN‑6.C (LO)

## Video transcript

so we want to evaluate the definite integral from negative 1 to 2 from negative 1 to negative 2 of 16 minus X to the third over X to the third DX now at first this might seem daunting I have this rational expression I have X's in the numerators and X's in the denominators but we just have to remember we said to do some algebraic manipulation and this is going to seem a lot more tractable this is the same thing as the definite integral from negative 1 to negative 2 of 16 over X to the 3rd minus X to the 3rd over X to the 3rd minus X to the 3rd over X to the 3rd DX and now what is that going to be equal to that is going to be equal to the definite integral from negative 1 to negative 2 of I could write this first term right over here let me do this in a different color I could write I could write this as 16 X to the negative 3 X to the negative 3 and this second one we have minus X to the 3rd over X to the 3rd well X to the 3rd is just over X to the 3rd over X to the 3rd is just going to be equal to 1 so this is going to be minus 1 DX so DX and so what is this going to be equal to well let's take the antiderivative of each of these parts and then we're going to have to evaluate them at the different bounds so let's see the antiderivative of 16 X to the negative 3 we're just going to do the power rule for derivatives in Reverse you could do this as the power rule of integration 4 or power rule of taking the antiderivative where what you do is you're going to increase our exponent by 1 so you already know from negative 3 to negative 2 and then you're going to divide by that amount by negative 2 so it's going to be 16 divided by negative 2 times X to the negative 2 all I did is I increase the exponent and I divided by that amount so that's the antiderivative here and 16 divided by negative 2 that is just negative 8 so we have negative 8 X to the negative 2 and then the antiderivative of negative one well that's just negative x negative negative x negative x and actually you could you might just know that and hey if I take the derivative of negative x I get negative 1 or if you view this as negative X to the 0 power because that's what 1 is well it's the same thing you increase the exponent by 1 to get X to the first power and then you divide by 1 and so I mean you could view it as that right over there but either way you get to negative or minus X and so now we want to evaluate that we're going to evaluate that at the bounds and take the difference so we're going to evaluate that at negative 2 and then subtract from that this evaluated at negative 1 and let me do those in two different colors so we can see what's going on so we're going to evaluate it at negative 2 and we're going to evaluate it at negative 1 so let's first evaluate it at negative 2 so this is going to be equal to this is going to be equal to when you evaluate it at negative 2 it's going to be negative 8 negative 8 times X to the negative 2 so negative 2 to the negative 2 power minus negative 2 and from that we're going to subtract it evaluated at negative 1 so it's going to be negative 8 times negative 1 to the negative 2 power minus negative 1 all right so what is this going to be so negative 2 to the negative 2 so negative 2 to the negative 2 is equal to 1 over negative 2 squared which is equal to 1/4 so this is equal to positive 1/4 but then negative 8 times positive 1/4 is going to be equal to negative 2 and then we have negative 2 minus negative 2 so that's negative 2 plus 2 and so this everything I've just done in this purplish color that is just going to be zero and then if we look at what's going on in the orange revaluated negative one let's see negative one to the negative 2 power well that's 1 over negative 1 squared well this is all just going to be 1 and so we're going to have negative 8 plus 1 which is equal to which is equal to negative 7 so all of this evaluates to negative 7 but remember we're subtracting negative 7 and so this is going to result we deserve a little bit of a drumroll this is going to be equal to positive positive 7 and obviously we don't have to write that positive out front I just wrote that just to emphasize that this is going to be a positive 7
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