A Tip

BigGlenntheHeavy

Senior Member
Joined
Mar 8, 2009
Messages
1,577
This is in regards to Aladdins query below.\displaystyle This \ is \ in \ regards \ to \ Aladdin's \ query \ below.

Hint: A tip, b a constant, obviously b  0.\displaystyle Hint: \ A \ tip, \ b \ a \ constant, \ obviously \ b \ \ne \ 0.

duu2b      Natural Logarithm Solution, 1u2b can be expanded.\displaystyle \int\frac{du}{u^{2}-b} \ \implies \ Natural \ Logarithm \ Solution, \ \frac{1}{u^{2}-b} \ can \ be \ expanded.

duu2+b      Arctangent Solution, 1u2+b is prime.\displaystyle \int\frac{du}{u^{2}+b} \ \implies \ Arctangent \ Solution, \ \frac{1}{u^{2}+b} \ is \ prime.
 
Anotherway

duu2+b2=i2blnu+ibuib+C\displaystyle \int\frac{du}{u^2+b^2} = \frac{i}{2b}ln\left |\frac{u+ib}{u-ib}\right |+ C

This representation is useful in complex analysis.
 
Yes, indeed. There are literally millions of these identities in some form or another.

Here's one, we do not see very often, they I like. The clever use of which may prove useful in solving various limits or what not.

sin(x)x=cos(x2)cos(x4)cos(x8)\displaystyle \frac{sin(x)}{x}=cos(\frac{x}{2})\cdot cos(\frac{x}{4})\cdot cos(\frac{x}{8})\cdot\cdot\cdot

sin(x)x=k=1cos(x2k)\displaystyle \frac{sin(x)}{x}=\prod_{k=1}^{\infty}cos(\frac{x}{2^{k}})

which means that limn2nsin(x2n)=x\displaystyle \lim_{n\to \infty}2^{n}sin(\frac{x}{2^{n}})=x

Here is one for arctan related to SK's, it would appear.

tan1(x)=i2ln(i+xix)\displaystyle tan^{-1}(x)=\frac{i}{2}\cdot ln(\frac{i+x}{i-x}).
 
Top