How do you find #int (2x^2 -x)/((x^2 -1)^2)dx# using partial fractions?
First let's factor the denominator:
we see the difference of squares, so we can break that down like so...
Now we can "distribute" the power to the two terms since they are being multiplied
Now we see two repeated linear terms, so our partial fraction decomposition will look like this:
After getting common denominators you should get the following (Sorry, I kinda skipped a step):
To find the constants, let's just plug something in for
We might need some other trickery to find
let's try multiplying the
This is a MESS to sort through, but a quick thing we can do is look at the coefficients in front of the powers on the left side...
These have to match the coefficients in front of the power on the right side, so we'll collect up only the coefficients in front of one at a time...
now let's do the constants...
since we know the following :
you should be able to figure out
now we can finally rewrite the integral like so:
These two parts should be easy...
Now the other two...
all together we have: