for example, how does
You know how the numerator must be a lower degree than the denominator and if it isn't then you need to use long division/synthetic division? That's were I'm stuck. For example I tried doing long division but then I got a remainder and I'm not really sure what I'm doing.
Also it kind of confuses me when it's not just one constant on the top (e.g. A/(x-9)) how does this work? For example, if one of the factors in the numerator is then would I have ?
. The integral of x is minus the integral of . The last can be done by "partial fractions": write it as . The first can be integrated by the substitition and the second can be integrated as an arctangent.
Surely you mean "one of the factors in the denominator is . In that case you can write .Also it kind of confuses me when it's not just one constant on the top (e.g. A/(x-9)) how does this work? For example, if one of the factors in the numerator is then would I have ?
the harder, but more straight forward way is to use long division or synthetic division of polynomials. it is really annoying to teach and/or type out such a solution here, so i refer you to google.
a nicer method, but one that requires more insight, is algebraic manipulation
this sight is very good for teaching long division with polynomials. http://www.sosmath.com/algebra/factor/fac01/fac01.html
with partial fractions it confuses me when you don't just have A on the top but Ax+B. does only
for example (34 sec in) the last term should be Dx+E/(1+x^2) right? that's because the x is squared? what would happen if the entire term was squared? also in the video I don't get where the A/x comes from, wouldn't it be A/x^2 ?
the idea is the numerator must be one degree less than the denominator. that's the basic principle, it does get a tad weird with awkward (non-factorable) polynomials though
where is 1 + x^2 coming from?
for example (34 sec in) the last term should be Dx+E/(1+x^2) right? that's because the x is squared? what would happen if the entire term was squared? also in the video I don't get where the A/x comes from, wouldn't it be A/x^2 ?
Thanks, finally I get the answer to
There's one question I still have: after doing long division I get where != denotes "not equal" relation. Could someone explain to me why the minus doesn't distrubute over to the second term from the partial fraction? like if it's minus something, and then that something becomes a partial fraction, why wouldn't the second fraction be negative?
the original question is integrate
the correct answer is definatly
this is what I did:
by division becomes so that can be integrated as
For the first integral I used u substition and let
and for the second integral I used trig identities after I factored out the 4
so then plugging everything back in I would have but then the negative makes the arctan negative and the answer is off by one sign
I found out what I was doing wrong.