Partial fractions decomposition. Why $cx+d$ instead of $cx$ for the numerator of $(x^2+2$). I understand that the aim of partial fractions decomp. is simply to reach (an) integrable functions, but then I have trouble wrapping my head around why you cannot make the numerator of something like $x^2+2$ equal to $Cx$ alone and then later use u-sub. The question I tried this on and failed was this:enter image description here


Of course you would try to look for $Cx$ in your case because in general to integrate $\frac 1{x^2+bx+c}$ where the denominator cannot be further factored in $\mathbb{R}$, you would aim for something like $\frac{C(2x+b)}{x^2+bx+c}$. In your case $b=0$ and hence you would look for $Cx$.
In your case your $Cx$ appears naturally:
$$\frac{x^3+5}{x^2(x^2+2)}= \frac{\overbrace{x}^{Cx}}{x^2+2} + \frac{5}{x^2(x^2+2)}$$
$$=\frac{x}{x^2+2} + \frac 52\frac{2+x^2-x^2}{x^2(x^2+2)}$$
$$=\frac{x}{x^2+2} + \frac 52\frac 1{x^2}- \frac 52\frac 1{x^2+2}$$
