# Integrals of Polynomials

# Basic Formula

We shall now prove that polynomials can be integrated using the following Basic Formula: If , then with , then we have

To prove this, we note that for we have

- ,

which is the same as

- .

To see this we note that if , then and thus .

For we are supposed to prove that

- ,

which is the same as . To see this we note that if , then

- .

If now is small then we can argue that is so small that it can be neglected, and thus .

For we are supposed to prove that

- ,

which is the same as

- .

To see this, we note that if $x(t)=t^3$, then

- .

If now is small then we can argue that and are so small that they can be neglected, and thus . Similarly, one can show the Basic Formula:

In MTS we show that his formula also holds for negative exponents

# The Logarithm

We shall also discover that the case is special, and gives rise to the logarithm

as the solution of

- ,

that is

- .

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