We are going to understand Problem reduction:
|----> [simpler problem part]
[problem] -- transform --(AND node)|----> [simpler problem part]
|----> [simpler problem part]
In the realm of integral calculus, there are all sorts of simple methods (transformations) that we can try, that will take a complex problem and make it into an easier problem.
Educational philosophy
- Skill
- Understand
- Widtness
You want to acquire a skill. First you need to understand it, and it is best done while witnessing it (through examples).
Example of Problem reduction with an integral
How do we go about integrating this:
∫[(-5x^4)/(1-x2)^(5/2)]dx
How can we simplify the above integral ?
IMPORTANT: we want to list the safe transformations (that work all the time no matter what). "trick substitution" is a heuristic approach that might not work all the time, so discarded.
SAFE transformations table
∫-f(x)dx = -∫f(x)dx- Take the constants out:
∫cf(x)dx = c∫f(x)dx - Sum of integrals is the integrals of the sum:
∫∑f(x)dx = ∑∫f(x)dx - Divide when the degree of the nominator is greater than that of the denominator:
∫P(x)/Q(x)dx -> divide
Solution Strategy
Devise a procedure, architecture, a framework to apply the knoledge that we are gaining
- Apply all safe transformations
- Look in table
- Test if done
- Call for success
Let's apply
∫[(-5x^4)/(1-x2)^(5/2)]dx <=>
# Apply 1.
∫[(5x^4)/(1-x2)^(5/2)]dx <=>
# Apply 2.
∫[(x^4)/(1-x2)^(5/2)]dx <=>
We can only go so far, with the safe transformations. So let's try to find som heuristing transformations.
Heuristic transformations
Heuristic: a funny word, that meanst the thing often works, but not guaranteed to work. It's an attempt.
f(sinxcosxtanxcotxsecxcscx) = g1(sinxcosx) = g2(tanxcscx) = g3(ctxsecx)