Generally speaking, how should I approach a problem involving factoring? I usually don't have a problem with the more typical forms, but sometimes I just don't know what to do.
My calc2 question is this:
The given curve is rotated about the y-axis. Find the area of the resulting surface.
$$y = \frac{1}{4}x^2- \frac{1}{2}\ln x,\quad 1 \le x \le 3$$
So the first thing I did was this:
$$\frac{dy}{dx}= \frac{1}{2}x-\frac{1}{2x}$$
And then this:
$$1+\left(\frac{dy}{dx}\right)^{\!2}= 1 + \frac{1}{4}x^2-\frac{1}{2x}+\frac{1}{4x}$$
I was then able to simplify it down to this:
$$\frac{x^2}{4}+\frac{1}{2}+\frac{1}{4x^2}$$
But leaving it in this form doesn't make it an easy integral to deal with. My solution manual shows them simplifying it to this:
$$\left(\frac{x}{2}+\frac{1}{2x}\right)^{\!2}$$
And I have no idea how to get from the form I was in to the form above. Is there any strategy, technique, books or videos I can read to strengthen my factoring ability?
Lastly, how did they get to that form above?
You have $$\frac{x^2}{4}+\frac{1}{2}+\frac{1}{4x^2} = {x^4 + 2x^2 + 1\over 4x^2}.$$ The denominator is a perfect square; now factor $${x^4 + 2x^2 + 1\over 4x^2} = {(x^2 + 1)^2\over 4x^2} = \left(x^2 + 1\over 2x\right)^2. $$
Does this make it less opaque?