Question

what is the partial decomposition of...
x^3+x^2-5/(x^2+7)^2

Answer 1

so the denominator is a square of an irreducible quadratic, so it has to separate to (Ax+B)/(x^2+7) + (Cx+D)/(x^2+7)^2. Then you have to multiply by the denominators to get what you are looking for, so \[x^3+x^2-5=(Ax+B)(x^2+7)+(Cx+D)\]\[x^3+x^2-5=x^3(A)+x^2(B)+x(7A+C)+(7B+D)\
so A=1, B=1, -7 = C and -12=D. by relating the coefficients on the left to the ones on the right. Then sub in your values andyou have your decomposition. and can integrate.