MHB How to Solve this Partial Fraction Decomposition Problem?

[shamieh]

$$\int \frac{7dx}{x(x^2+8)^2}$$

so I am thinking its going to be set up like: $$\frac{A}{x} + \frac{Bx + C}{x^2 + 8} + \frac{Dx + E}{(x^2 + 8)^2}$$
Practice problem I'm stuck on.
so I cleared fractions and got A = 7/64 , b = -7/64 and C = 105/64 and now I'm lost... can anyone work this problem for me so I can see what's going on? Thanks in advance.

 

[Sudharaka]

$$\int \frac{7dx}{x(x^2+8)^2}$$

so I am thinking its going to be set up like: $$\frac{A}{x} + \frac{Bx + C}{x^2 + 8} + \frac{Dx + E}{(x^2 + 8)^2}$$
Practice problem I'm stuck on.
so I cleared fractions and got A = 7/64 , b = -7/64 and C = 105/64 and now I'm lost... can anyone work this problem for me so I can see what's going on? Thanks in advance.

Hi shamieh, :)

I think your value for C is not correct. However the other two values are correct.

\[\frac{7dx}{x(x^2+8)^2}=\frac{A}{x} + \frac{Bx + C}{x^2 + 8} + \frac{Dx + E}{(x^2 + 8)^2}\]

\[\Rightarrow A(x^2+8)^2+(Bx+C)(x^2+8)x+(Dx+E)x=7\]

Substituting different values for $x$ you can get three simultaneous equations (assuming you already found the values A and B correctly) which can be used to find the remaining unknowns, $C,\,D\mbox{ and }E$.