How do you use partial fraction decomposition to decompose the fraction to integrate #(x^2+x+1)/(1-x^2)#?
Divide first, then find the partial fraction decomposition.
Before looking for a partial fraction decomposition, we must have the degree of the denominator strictly less than that of the numerator.
So we need to divide or regroup to get:
# = (x^2-1)/(1-x^2)+(x+2)/(1-x^2)#
# = -1 - (x+2)/(x^2-1)#
Now we can get the partial fraction decomposition for:
Putting it all together we have:
# = -1 - (3/2)/(x-1) + (1/2)/(x+1)#