Indefinte integrals

Question

My daughter is having problems with this one and it's way above me. Any help is appreciated.

Consider the indefinite integral ((9x^3+8x^2+45x+54)/x^4+9x^2)dx

then the integrand has partial fraction decomposition (a/x^2)+(b/x)+(cx+d/x^2+9) where

a=?

b=?

c=?

d=?

Integrating term by term, we obtain that ((9x^3+8x^2+45x+54)/x^4+9x^2)dx= ?

She is supposed to be able to have solutions for every spot there is a "?". Can anyone show me/her how to work this?

Answer 1

Some of your parentheses are out out place. The correct expression is:

(9x^3+8x^2+45x+54)/(x^4+9x^2) = (a/x^2)+(b/x)+(cx+d)/(x^2+9)

As with any Partial Fraction problem, multiply both sides by the factored form of the denominator on the left, which is x^2(x^2+9). After distributing and cancelling, this clears the denominators on the right:

(9x^3+8x^2+45x+54) = a(x^2+9) +bx(x^2+9) + (cx + d)x^2 (**)

There are two ways to go from here. You can expand the right side and equate coefficients of powers of x:

1. x^3 : 9 = b + c
2. ...
3. const: 54 = 9a

which gives four equations for a, b, c, d. Alternatively, you can choose four different values of x and plug into the equation (**). x=0 is the only obvious choice here, since that gives 54 = 9a.

Either way, the result is a 4x4 system of linear equations in a,b,c, and d, which you can solve by any convenient method (elimination, substitution, Gauss Elimination). That will take a bit of work.

Answer 2

... the partial fractions will take the form (Ax+B)/x^2+ (Cx+D)/(x-3)^2