Laplace Transform of x^2 + e^x

Yes, readers who understand a bit of calculus are going to have an unfair advantage during today's blog post. Hey, I make the rules, I can use calculus if I like. Besides, calculus is the key to life (also, 42 is the answer to Life, the Universe, and Everything.)

Roughly speaking, calculus can be divided into two general disciplines.

• Integration - calculating the area below curves
• Differentiation - finding the slope of curves

As the advanced mathematician can attest, either of these operations is vastly more difficult to do than their algebraic underlings. (how I cry for the days of multiply, divide, add, and subtract.)

When you get really crazy and deep into the mathematical curricula, you learn about something called the Laplace transform. It's a handy way to analyze mechanical, optical, and electronic systems, but one of its niftiest side-effects is what it does to the basic operations of calculus.

Simply put, differentiation and integration become multiplication and division, respectively. Take the Laplace transform of a function, divide it by s, then find the inverse Laplace transform, and voila, you've integrated. Ditto for multiplication.

All this to say, I wish there were a Laplace transform for life, but alas, there doesn't appear to be. Let the alchemists have the gold; give Indy the holy grail; all I want is a coherent way to unravel life's mysteries, is that too much to ask? (no, 42 doesn't check out... its derivative is zero and its integral over the entire real line is infinity. Bo-orring.)

And for the record, the Laplace transform of x^2 + e^x is 2/s^3 + 1/(s - 1). It may come in useful some day. You never know. (I take check, Visa, PayPal...)