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)`