Intermediate step for integration? Among the various method of Integration there is one specific method(it may vary according to the terms) where for instance if we have a function as:$$\frac{px+q}{ax^2+bx+c}$$
To integrate this we perform the following intermediate step:
Let $$px+q=A\left(\frac{d}{dx}(ax^2+bx+c)\right)+B$$
And then proceed further by obtaining the values of $A$ and $B$. During the process of deducing $A$ and $B$ we equate the coefficients of similar degrees of variables and the constants separately.
In this case:
$$p=2aA$$ and $$q=Ab+B$$
I didn't understand this part. How does the business of equating the coefficients separately, work? Normally we don't do that for instance if we have a quadratic as:$$ux^2+vx+w=ax^2+bx+c$$
We don't say that $u=a,v=b$ and $w=c$. Do we? Instead don't we re-frame into the following quadratic equation:
$$\left(u-a\right)x^2+\left(v-b\right)x+\left(w-c\right)=0$$
How does that work? Is it some intrinsic property of functions?
 A: When we consider $x$ to be a solution of an equation, we are looking for $x$. But in this case we are not looking for $x$, we are looking for another way to write a function in $x$. Thus the two expressions must be equal for all $x$.
We can equate the coefficients because the powers of $x$ are linearly independent over the space of polynomials.
A: You want
$$
px+q = A(2ax+b)+B = (2Aa)x + (Ab+B) \text{ for ALL values of $x$}.
$$
The only way the quantifier "for ALL values of $x$" can be true is that the coefficients of corresponding terms of the polynomial are equal.  That means $p=2Aa$ and $q=Ab+B$.  The point is to find the values of $A$ and $B$ that make that true.  Then you can write
$$
\int\frac{px+q}{ax^2+bx+c}\,dx = \int \frac{A\, du}{u} + \int\frac{B}{ax^2+bx+c}\,dx
$$
The first integral becomes $\log|u|+\text{constant}$.
The fate of the second depends on whether the discriminant $b^2-4ac$ is positive, negative, or zero, and the first step toward evaluating the integral is completing the square.
