*Integration is a content-free business organizational model. This is to be confused with what this article is actually about, which is a mathematical theory.*

**Integration** refers to the branch of mathematics involved in finding the **Integral** of a function, which is defined formally by the following axiom:

where $ \Im $ is the set of incomprehensibly complicated integration techniques that your professor is unable to figure out in lecture, but nonetheless expects you to know for your final.

Thus, a more informal and traditional definition that suffices in most applications is

Applications of integration are diverse. By this, it is meant that its uses involve finding areas, volumes, and a vast array of other completely irrelevant quantities, though in many cases these are impossible to find because they hide so well inside those goddamn square roots. Naturally, none of these applications have any relevance to the real world, since nobody gives a damn about finding the distance travelled by an ant along the vector curve $ <ae^t,be^t>: t=0..1 $.

(in case you do give a damn, it's $ (e - 1)\sqrt{a^2 + b^2} $)

## HistoryEdit

Though integration has existed since the beginning of time, integrals have not. The reason for this is that integration is so completely impractical that it was never used -- and so nobody realized that it required the existence of integrals -- until some idiot received brain damage from an apple that fell on his head, resulting in a spurt of insanity which served as his inspiration for inventing the equally insane idea of the integral.

Since then, many people suffering from similar psychopathologies (sometimes colloquially known as mathematicians) have adopted and worked hard to expand on the idea of the integral to make its use even more academic and less practical (contrary to the predictions of many philosophers who thought that this was logically impossible), as well as to ensure that integration is kept sufficiently impossible to understand for first year college students.

## The Mental Theorem of IntegrationEdit

This is the theorem upon which all integration is based on. Essencially, it states that the integral of a function $ f(x) $ is practically impossible to find. It was formerly called the Fundamental Theorem of Integration, but in recent years, it has come to be called the Mental Theorem to better reflect the deterioration of mental health in those that seek to understand it.

Formally, the theorem states that:

If

then

where

such that

**Proof** *(by contradiction)*:

Assume that

$ \int{f(x)} \not= [\diamondsuit \searrow \heartsuit \to \clubsuit \leftarrow \spadesuit] $

Then we have:

$ \int{f(x)} $

$ = a $ where $ a \not= [\diamondsuit \searrow \heartsuit \to \clubsuit \leftarrow \spadesuit] $

$ = qg $ [by theorem 209.3 v.4]

$ = \frac{qg}{f_{27}} - f_{26} $ [by the Llama lemma]

$ = \int{(\frac{\int{\frac{ \int{\frac{ \int{\frac{\int{\frac{ }{ } } }{ \int{\frac{ }{ } }} }}{ \int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} }} }}{\int{\frac{\int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} } }{ \int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} }} } } } }{\int{\frac{ \int{\frac{\int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} } }{ \int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} }} }}{ \int{\frac{ \int{\frac{ \int{\frac{ }{ } }}{ \int{\frac{ }{ } }} }}{ \int{\frac{ \int{\frac{ }{ } }}{\int{\frac{ }{ } } } }} }} } } )} f_{81271279}^{\pi} - 483 \int{ \frac {krt}{lol}} + 1337 - e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} $ [this is a trivial step]

$ = 1 $ [OMG this is taking forever. I quit.]

...

Clearly, since the proof is not completed, and yet we promised you a proof, then we are contradicting ourselves. Therefore by contradiction, our original supposition that

$ \int{f(x)} \not= [\diamondsuit \searrow \heartsuit \to \clubsuit \leftarrow \spadesuit] $

must be false. Therefore,

$ \int{f(x)} = I = [\diamondsuit \searrow \heartsuit \to \clubsuit \leftarrow \spadesuit] $

**Q.E.D.**

## List of Other Useful TheoremsEdit

## List of Other Useless TheoremsEdit

What follows is an (incomplete) list of integration theorems that can be mathematically proven to lack any sense of purpouse. All of these can be proven quite simply by the classical *method of exhaustion* in which one engages in extreme physical exertion to the point that the brain cannot think logically, thus accepting the validity of these theorems.

**Comparison Theorem**: If $ f = \int{} $, nothing compares to the awesomeness of $ f $.**Linear Approximation Theorem**: The mathematical notation for the integral, $ \int{} $, looks approximately like a squiggly line.**Differential Equation Theorem**: If $ f(x) = \int{x} dx $ and $ g(x) = \int{x} dx $, it is hard to tell the difference between f and g.**Improper Integral Theorem**: If $ f $ is the integral of $ x $, it is not proper form to write $ f(x) = \int{x} $.**Solution Theorem**: An ideal solution to any given problem $ f(x) = \int{x}dx = 0 $ is one that satisfies both parties in a nonagressive and mutually respectful manner.**Curve Sketching Theorem**: It is tricky to get the curve on the $ \int{} $ sign just right. So unless your $ \int{} $ looks like a $ Q $, don't worry too much.**Parametric Function Theorem**: If your $ \int{} $'s suddenly look like they're short of breath, it could be the result of an acute pencil stroke, so be sure to call a Parametric ASAP.

## Integrals in The Modern WorldEdit

There is a general consensus in today's academic community that an analogy can be drawn between the integral, $ \int{} $, and homosexuality. Both are inexplicable, not straight, contain 2 e's and a's each, and are often what nerds turn to when they can't get any real action. Such similarities are uncanny and have initiated a significant amount of research on the subject. Although preliminary findings suggest that that the scientific hypotheses behind these expriments are ludicrous and laughable and contain absolutely no merit whatsoever, what will ulitmately be concluded of such research remains to be seen.