Let's say,
$\int\frac{x+11}{x^2+8x+15}dx = \int\frac{x+4}{x^2+8x+15}dx+\int\frac{7}{(x+3)(x+5)}dx$
So can I use, for example,
$u = x^2+8x+15$ for $\frac{x+4}{x^2+8x+15}$
and
$v=x+3$ for $\frac{7}{(x+3)(x+5)}$
It's like I'm solving them separately with different substitutions although they are actually parts of the bigger equation. Is this possible or do I have to stick with the same substitution under the same equation?
I know about the double substitution method but I just want to explore more ways to tackle such questions.
Better is to write $$\frac{x+11}{x^2+8x+15}=\frac{4}{x+3}-\frac{3}{x+5}$$