Experiment with a Constant of Integration
When we integrate a function, we generate an arbitrary constant of integration, C.
(x²)/2 + C = ∫ x dx
This C can be any real number.
Today, we will conduct and experiment. Let us start with some simple value, 1. Let us pick a simple constant of integration, C = 1. Now we start integrating.
∫ 1 dx = x + C = x + 1
OK, Let us repeat.
∫ x + 1 dx = x²/2 + x + 1
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OK, again.
∫ x²/2 + x + 1 dx = x³/6 + x²/2 + x + 1
Interesting. That looks familiar. Lets reverse it.
1 + x + x²/2 + x³/6
This is the beginning of the series for e^x, nice.
m
∑ x^n/n!
n=1If we keep going, we get more terms for e^x.
It seems like e^x is born ex nihilo. But actually, e^x is growing from the seed we planted, f(x) = 1. The fertilizer we feed it, C = 1. And we water with the recursive application of integration.
Since we keep integrating and adding a 1, we are reversing the process of differentiation of e^x, which removes the 1, and lowers the powers. The full infinite series for e^x is infinitely differentiable. We can integrate the infinite series an infinite number of times as well.
