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 sin2x in the denominator..
2. Factoring using the +C -C trick
INTEGRAL( dx / (x2 + 10x + 30) )
You have to factor (wactor! as my friend used to say) the denominator into ((x+5)2 + 5) by:
x2 + 10x + 25 - 25 + 30 = (x + 5)2 + 5
You know to do +25, -25 BECAUSE half of 10 is 5, and 52 is 25. That method in general is called completing the square and you have to do it all the time.
