Well, this post was looking for the same thing I was, and nobody seemed to know of any.

The problem with algebra tricks is they’re so context dependent. There’s really only a few things that math books and tests use to make “wicked” problems (list being expanded).

- 1. Multiplying by the conjugate
- 2. Factoring using the +C -C trick (complete the square method)

Really the “hard” problems require an algebraic twist, a keen eye, insight – stuff you only get after having practiced with hundreds of problems (and checking the solutions!).

So first,

sosmath: Derivative of sin, cos, tan, csc, sec, cot

Mathworld: Inverse trig functions

Now, the rest of this post is just examples of the trick (by name) and a couple of examples that illustrate the trick in action (list being expanded!) (Bear with the non-mathmlness for now!)

# 1. Multiplying by the conjugate

INTEGRAL( dx / (1 + cos x) )

To solve, you have to multiply the numerator and denominator by ( 1 – cos x ) to get sin^{2}x in the denominator..

# 2. Factoring using the +C -C trick

INTEGRAL( dx / (x^{2} + 10x + 30) )

You have to factor (wactor! as my friend used to say) the denominator into ((x+5)^{2} + 5) by:

x^{2}+ 10x + 25 - 25 + 30 = (x + 5)^{2}+ 5

You know to do +25, -25 BECAUSE half of 10 is 5, and 5^{2} is 25. That method in general is called completing the square and you have to do it all the time.