# Integration by parts, DI method

(Traditional Method learnt at O-Level) Integration by Parts:

NEW Technique taught by this UC-Berkley Chinese Mathematician: “DI” Method

First Stop: “D column at 0

2nd Stop:

3rd Stop:

$\displaystyle \int {e^{x}}\sin {x} dx$

# e^pi vs. pi^e

Prove:

$\boxed{e^{\pi} > \pi^{e}}$

# A Brilliant Limit

$\displaystyle \boxed{ \lim_{n \to \infty}\left({\frac{n!}{n^n}}\right)^{\frac{1} {n}} = \frac{1} {e}}$

Method:

Note:

See a similar limit question, but by the more theoretical rigorous French pedagogy :

# Recursive Technique

Recursive Technique is a Math trick which is fun for math competitions, but not much mathematical value.

$f(x) = x^{x^{x^{... } } }$

$g(x) = \sqrt{x{\sqrt{x{\sqrt{x...}}}}}$

You can integrate or differentiate f(x), or g(x) by using this recursive definition:

Let

$y= x^{x^{x^{... } } }$
$\implies f(x) = x^{y }$

$z = \sqrt{x{\sqrt{x{\sqrt{x...}}}}}$
$\implies (x) = \sqrt{x{z}}$

# Integral of x^x from 0 to 1

$\boxed {\displaystyle\int_{0}^{1}{x^{x}dx }= \frac {1}{1^1} - \frac {1}{2^2} +\frac {1}{3^3} - \frac {1}{4^4} + \frac {1}{5^5} +...}$

$\boxed {\displaystyle\int_{0}^{1}{x^{-x}dx }= \frac {1}{1^1} +\frac {1}{2^2} +\frac {1}{3^3} + \frac {1}{4^4} + \frac {1}{5^5} + ...}$

Recall: Gamma function

$\boxed{\displaystyle \Gamma(z) = \int_{0}^{\infty}{ t^{(z-1)}.e^{-t}dt }}$

$\boxed {\Gamma (n+1) =\int_{0} ^{\infty}{t^{n}. e^{-t}dt}= n!}$