So, partial fractions were the main focus this week, along with sharpening our integration skills. Some things you need to remember when it comes to partial fractions, is that there are basically 4 cases, including synthetic division, that you can use when regular substitution fails, and by parts I think too. So, basically, when you see a fraction...
1. See if it can be solved using synthetic division (i.e. degree of top greater than degree of bottom)
2. If synthetic won't work, then Factor the Bottom!
Once you're dong factoring, you're gonna want to split it up with different letters over all the factors...so A B C, and so on.
However. If you have say a (x+1)^2, you have to put that particular factor over Cx+D...don't ask why, it's just the way of the gods..
So once you decide what to do after factoring, you go through and plug in numbers for x (after finding common denominators) that cancel out the other letters, till you have values for all letters (variables) used.
Once finding those values you go BACK, again, and plug those in in the appropriate places. Once you get that, then you FINALLY integrate. Yayy!! finally! (Most of the times you should get a lot of natural logs...just saying. Got it? good.
Okay...a question...what the heck is this integration table thing?? Unfortunately I was absent that day, so I was unable to get a good explanation....so if someone could do a problem or two and explain each step, it'd be much appreciated....:DD Thanks!!!
Sunday, September 19, 2010
post 4
sorrryyyy i missed post 3. i fell asleep before i did it, haha. anyways...
this week in calc we learned partial fractions and went over more trig sub! yayyyy. and we learned about integration tables on friday. which is pretty cool that all you have to do is plug it into a formula.
partial fractions is used when you pretty much have no other choice. and it has to be fraction. and your top degree is smaller then your bottom degree.
becuase when your top degree is larger than your bottom degree, you would use synthetic division, which may i say makes life so easy. haha
anyways, here's how you do partial fractions...
1. you factor the bottom.
2. you do a/first term, b/second term, c/third term..etc & set that equal to your equation****
3. you then do common denominator and get rid of all the fractions.
4. you plug in CONVENIENT x values.
5. solve for a, b, c, d and what not.
6. then you go back to step 2 and plug in all the numbers you got.
7. integrate & solve
****now there is a tricky part for step two. if you have something squared when you factor. like x^5/(x+1)^2
you would do this
a/(x+1) + b/(x+1)^2
... and if it was cubed, or to the fourth or whatever, you would go all the way up to that degree.
ALSO, if you have something inside of it squared after factoring.. like this
x^5/x(x^2+1)
then you would put the term that has a squared in it like this...
a/x + (bx+c)/(x^2+1)
and you would do that for whatever letter you are at... for example if you already had an a & b, you would do cx + d/ whatever.
get it? k good :)
can someone go over synthetic division again. for some reason i keep forgetting it.
this week in calc we learned partial fractions and went over more trig sub! yayyyy. and we learned about integration tables on friday. which is pretty cool that all you have to do is plug it into a formula.
partial fractions is used when you pretty much have no other choice. and it has to be fraction. and your top degree is smaller then your bottom degree.
becuase when your top degree is larger than your bottom degree, you would use synthetic division, which may i say makes life so easy. haha
anyways, here's how you do partial fractions...
1. you factor the bottom.
2. you do a/first term, b/second term, c/third term..etc & set that equal to your equation****
3. you then do common denominator and get rid of all the fractions.
4. you plug in CONVENIENT x values.
5. solve for a, b, c, d and what not.
6. then you go back to step 2 and plug in all the numbers you got.
7. integrate & solve
****now there is a tricky part for step two. if you have something squared when you factor. like x^5/(x+1)^2
you would do this
a/(x+1) + b/(x+1)^2
... and if it was cubed, or to the fourth or whatever, you would go all the way up to that degree.
ALSO, if you have something inside of it squared after factoring.. like this
x^5/x(x^2+1)
then you would put the term that has a squared in it like this...
a/x + (bx+c)/(x^2+1)
and you would do that for whatever letter you are at... for example if you already had an a & b, you would do cx + d/ whatever.
get it? k good :)
can someone go over synthetic division again. for some reason i keep forgetting it.
Week Number Four
This week we reviewed trig. substitution integration and we also learned how to solve integration using partial fractions.
You can actually use partial fractions to break up fractions even if you aren't using it for integration.
Some simple steps for partial fraction integration:
1) First of all, make sure it isn't any other type of integration.
2) If you can factor the top or bottom then do so (if you can cancel anything then do so).
3) Once you've ruled out everything else, you then split the bottom factors into A/(factor1) + B/(factor2) +... = original.
4) Get common denominators and add.
5) Pick a convenient value for x (one that would give you zero once plugged into a factor) and plug in.
6) Solve for A, B, ... .
7) You then take the values of your variables and plug back in to when you first broke up the fraction.
8)Integrate! It will almost always be natural log integration. Remember that any number in front of a natural log is also it's exponent.
Question:
For integration using charts like we did in class on friday, how exactly do I know which formula to plug in to if the problem I'm facing doesn't have all of the necessary components. I.e. if the formula has an x and my problem doesn't.
You can actually use partial fractions to break up fractions even if you aren't using it for integration.
Some simple steps for partial fraction integration:
1) First of all, make sure it isn't any other type of integration.
2) If you can factor the top or bottom then do so (if you can cancel anything then do so).
3) Once you've ruled out everything else, you then split the bottom factors into A/(factor1) + B/(factor2) +... = original.
4) Get common denominators and add.
5) Pick a convenient value for x (one that would give you zero once plugged into a factor) and plug in.
6) Solve for A, B, ... .
7) You then take the values of your variables and plug back in to when you first broke up the fraction.
8)Integrate! It will almost always be natural log integration. Remember that any number in front of a natural log is also it's exponent.
Question:
For integration using charts like we did in class on friday, how exactly do I know which formula to plug in to if the problem I'm facing doesn't have all of the necessary components. I.e. if the formula has an x and my problem doesn't.
Post #4
This week in Calculus, we learned partial fractions and how to use a table to integrate easier. I think partial fractions are pretty easy, but that's being said with fingers crossed! So, let me try to explain what i don't understand. How do you know what part of the equation to use to find the right equation in the table?
So, let me explain partial fractions.
1. How do you know its a partial fraction?
Well duhhh sillies, its going to have a fraction with some quadratics
2. So, then what do you do? You need to break the fraction up.
3. You break up the fraction, by factoring the bottom and rewriting it as multiple new fractions.
4. Take the separate denominators with a numerator of A, B, C, or D.
5. Next, you create a common denominator and set that equal to the numerator of the initial fraction.
6. You then pick convient values to solve for the variables of the equation.
7. After you find the values, you plug everything back in to the fractions you created
8. Integrate
9. And hopefully box off the correct answerrrr!
So, let me explain partial fractions.
1. How do you know its a partial fraction?
Well duhhh sillies, its going to have a fraction with some quadratics
2. So, then what do you do? You need to break the fraction up.
3. You break up the fraction, by factoring the bottom and rewriting it as multiple new fractions.
4. Take the separate denominators with a numerator of A, B, C, or D.
5. Next, you create a common denominator and set that equal to the numerator of the initial fraction.
6. You then pick convient values to solve for the variables of the equation.
7. After you find the values, you plug everything back in to the fractions you created
8. Integrate
9. And hopefully box off the correct answerrrr!
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