in this video we're Computing the integral of an ugly rational expression and the natural tool to
use here is partial fractions decomposition now when we propose a partial fractions decomposition
the idea is that a proper fraction can be broken down into a sum of a bunch of other
proper fractions that are smaller and what I mean by proper fraction is the degree of the numerator
must be less than the degree of the denominator so the rational expression we're looking at
here has a problem we need to knock down that degree and that's when we start with polynomial
long division so we're going to take X cubed plus 2 and divide by x squared plus 2X Plus 1.
note that we're going to need some placeholders
we need a 0x squared in here we need a 0x and then we have our plus 2 at the end and then we start to
guess the thing that we multiply x squared plus 2X plus 1 by to get this polynomial at the bottom
plus possibly a remainder so we'll start with an x and distribute that over x squared plus 2X Plus 1.
then we find out how much we missed by in other words we subtract the X cubes cancel here and I
end up with a negative 2x squared and a Negative X and then I have a 2. Now we move to the constant
part of our guess and that would be a negative 2 so I can match that negative 2x squared in
the leading term and I get a minus 4X and minus 2 as well and again we subtract the negative 2x
squareds cancel negative x minus negative 4X is positive 3x and 2 minus negative 2 is plus 4. and
remember the interpretation of remainder here is that we're left with a 3X plus 4 divided by
x squared plus 2X plus 1. all right so our integrand can be represented as x minus two
plus this fraction and now we're going to factor the denominator of that fraction and propose
a partial fraction's decomposition so we could write that fraction as 3x plus 4 divided by the
factored denominator and this denominator looks like it factors into X Plus 1 times X plus one in
other words X Plus 1 all squared recalling the partial fractions decomposition rules so these
rules are set up so that the partial fraction is decomposition is guaranteed to work I need
a term that looks like a constant over X Plus 1 in addition to a term that's a constant over X
plus one all squared now we can clean things up by multiplying both sides by the least common
denominator which is X Plus 1 squared and this leaves us with a 3X plus 4 on the left hand side
and on the right hand side when we distribute X Plus 1 squared to the a containing term one factor
of X plus one is going to cancel but one survives and I end up with the a times the quantity X plus
one in the B containing term the denominator cancels that X Plus 1 squared completely and
I'm just left with a b and this equation isn't too hard to solve by just collecting the coefficients
of the different powers of X so on the left hand side we have three X plus four on the
right hand side I have an ax that's the only X containing term and that means a is equal to 3.
I also have a constant and that's just a plus b so again we compare the coefficients of like
terms and I end up with a equals three then when I compare the constants I end up with 4 equals
a plus b but I just said a is 3 and that means B is equal to one so now that we found a and b
we can replace this rational expression part in the remainder with our partial fractions
decomposition don't forget to keep the x minus 2 out in front so let's rewrite the integral
and now each of these pieces can be integrated with a simple guess so I have applying the power
rule to the X I have one half x squared the negative 2 integrates to negative 2x the 3
over X plus one that's a natural log type of antiderivative the 3 is a spectator here and
I have the natural log of the absolute value of x plus one and one over X plus one quantity
squared we have to view that as X Plus 1 to the negative 2 and that integrates to X Plus 1
to the negative 1 divided by negative one well that just puts a minus sign out in front and X
Plus 1 to the negative 1 that's a one over X plus one don't forget to add the plus c
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