06/28/2010, 08:16 PM

Hello!

I tried integrating the function x^^n for any nonnegative integer n. I wanted to share my method and results, so that you can check fofr accuracy (if I've done it correctly, it'd be a rare occurance - I haven't taken calculus as a class yet, so I went off of what I've read in books).

First we write

Then we can expand on the indefinite integral in the parentheses.

Once we have calculated this out, we can use substitution.

If we repeat this process n-1 times, then we get

Notice that I defined i_0 = 1. This merely simplifies the notation.

I haven't yet completed that general integral, but what matters is that it can be completed on a case-by-case basis through integration by parts. This is demonstrated for n=2...

Integrating by parts,

If we do this indefinitely, we end up with

Plugging in,

Bernoulli's integral, the so-called "Sophomore's Dream", is calculated as

Note that 0^0 is 1, so the second summation only has a non-zero value when n=k.

If anyone has any corrections, or if this has been done before, I'd love to know. Also, it'd be cool to see graphs of these functions from anyone who has Maple or Mathematica (or other amazing program).

Edit: Ugh. Just noticed that in the last few equations, I wrote the summations wrong. Rest assured, it's from k=0 to infinity.

I tried integrating the function x^^n for any nonnegative integer n. I wanted to share my method and results, so that you can check fofr accuracy (if I've done it correctly, it'd be a rare occurance - I haven't taken calculus as a class yet, so I went off of what I've read in books).

First we write

Then we can expand on the indefinite integral in the parentheses.

Once we have calculated this out, we can use substitution.

If we repeat this process n-1 times, then we get

Notice that I defined i_0 = 1. This merely simplifies the notation.

I haven't yet completed that general integral, but what matters is that it can be completed on a case-by-case basis through integration by parts. This is demonstrated for n=2...

Integrating by parts,

If we do this indefinitely, we end up with

Plugging in,

Bernoulli's integral, the so-called "Sophomore's Dream", is calculated as

Note that 0^0 is 1, so the second summation only has a non-zero value when n=k.

If anyone has any corrections, or if this has been done before, I'd love to know. Also, it'd be cool to see graphs of these functions from anyone who has Maple or Mathematica (or other amazing program).

Edit: Ugh. Just noticed that in the last few equations, I wrote the summations wrong. Rest assured, it's from k=0 to infinity.