Integrating via partial fractions My question is this:
Why is it that fractions have to be split up in a very specific manner?
For example if I have $\frac{5x}{(x+1)^2}$        this fraction HAS to be split up like this:$$\frac A{x+1} + \frac B{(x+1)^2}$$
And if it were $\frac{5x}{(x^2+1)^2}$ then it is split up in this way:
$$\frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2}$$
So why does it have to follow this very specific pattern? I can't find any information which is relevant on the internet. Thanks.

When doing partial fraction decomposition, each denominator in the sum $\sum_{i=1}^{n}\frac{p_i(x)}{q_i(x)}$ must be a factor of the original denominator of your function $f(x) = \frac{p(x)}{q(x)}$ (in lowest terms), so that when we write the sum with the lowerst common denominator
$$f(x) =\frac{\sum_{i=1}^{n}p_i(x)\prod_{j\neq i}q_j(x)}{q_1(x)\ldots q_n(x)}$$
we have that the resulting denominator $q_1\ldots q_n$ is divisible by the original denominator $q$. If our new denominator did not divide the original one, then it could not possibly be equal to $f(x)$ (for $p/q$ and our sum would have different denominators in lowest terms).
It is possible to use extra fraction terms whose denominators are not factors of the original denominator $q(x)$, but they will always cancel out in the end and so it is sufficient to ignore them.
