Der Anfang

· mathematics

A few times in the past, I’ve tried to start a blog, and I’ve never followed through with the discipline to post frequently and consistently.

This time, I hope, I shall remain active.

WordPress makes the inclusion of mathematical content easy, and so this blog will serve for any content that I might wish to write.

As an initial test, I offer the trivial derivation of the formula for integration by parts:

We begin by considering two functions u and v. Each function maps from from the reals to the reals. Consider a real number x, and consider the product u(x)\,v(x). The differential of the product is

\text{d}[u(x)\,v(x)] = v(x) \, \text{d} [u(x)] + u(x) \, \text{d} [v(x)]

\text{d}[u(x)\,v(x)] = v(x) \, u'(x) \, \text{d}x + u(x) \, v'(x) \, \text{d}x.

Solving for one of the terms on the right, we find

u(x) \, v'(x) \, \text{d}x = \text{d}[u(x)\,v(x)] - v(x) \, u'(x) \, \text{d}x.

Integrating both sides leads directly to the desired result

\int_a^b u(x) \, v'(x) \, \text{d}x = u(b)\,v(b) - u(a)\, v(a) - \int_a^b v(x) \, u'(x) \, \text{d}x.

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