What's the difference between 'constant of integration' and 'undetermined coeffcient'? I'm teaching myself Calculus. On page 573 of Thomas Calculus, it's said "Hence we have ... where K is the arbitrary constant of integration(to avoid confusion with the undetermined coefficient we labeled as C)." as the picture below shows.
This avoiding-confusion statement really confused me. Always, I never thought about the part attached at the end of indefinite integration. And, also it seems the author choose the letter C and K randomly in the Section 8.3 of this book.
So, can anyone tell me what's the difference between those two?
Any help will be appreciated. Thanks!
8.3 Integration of Rational Functions by Partial Fractions
 A: "Undetermined coefficient" just means a coefficient that we don't know yet, as usual in algebra.  The constant of integration $C$ is an unknown that arises in a certain context - it is the unknown constant that is necessary because there are multiple possible antiderivatives for any given function $f(x)$. It is usually denoted $C$, but in this particular case the symbol $C$ was already used with a different meaning, so the author chose to use $K$ instead.
A: $C$ and $K$ are just tokens.   The symbols have no fixed meaning until we assign them one.   So while $C$ is commonly used as an arbitrary constant from integration, if it is already used elsewhere then we need to assign a free symbol to that purpose.   $K$ is often a good standby, but we don't have to use either.
$$\int (Ux+V)\operatorname d x = \tfrac 12 Ux^2+Vx + T \qquad\text{where }{T\text{ is an arbitrary constant of integration}\\U,V\text{ are our unknown coefficients}}$$
A: In setting up a "decomposition template" for decomposing a rational function into partial fractions, it's conventional to go through as much of the alphabet as necessary for the "unknowns" in the numerators: those are the "undetermined coefficients" (although, I guess, once you figure out what they have to be, I guess they're "finally determined" coefficients) For example, writing
$$\dfrac{2x^2 + 5x - 1}{(x - 1)^2(x^2 + 3)} = \dfrac{A}{x - 1} + \dfrac{B}{(x - 1)^2} + \dfrac{Cx + D}{x^2 + 3},$$
the $A, B, C$, and $D$ are the "undetermined coefficients." 
It would be bad form to use $C$ to mean both "the coefficient of $x$ in the numerator of $\frac{Cx + D}{x^2 + 3}$" and "the constant of integration in $F(x) + C$"; those are two totally different things. So, whenever "$C$" (or anything, really) has already been used to name something, it's a good idea to not reuse the name for a different thing, at least within a single problem.
Hence the author choosing between ${}+C$ and ${}+K$ for constants of integration. I'll bet that if you look, the ${}+K$ is used only if the decomposition template already used $C$.
