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[Karlito]

That pi looked an awful lot like an 'n' at first.

[lemon10]

I am going to have to second wolfram alpha.
It solves pretty much every problem you can throw at it (as long as it understands the question at least). And it shows about a dozen other super useful things that you might want to know as well about a number or a function or whatever.
It even has a function where you it shows you step by step how it solves the problem. You can only do this 3 times per day in the free version though (although it only costs 3 dollars per month for the pro version as a student).

[Tarqiup Inua]

Why don't you just search for "why natural log" or something.

It's not thaaaat hard to find things.

Because I know what natural log is, I know what it is used for... I don't understand it and I cannot naturally use it.

I was hoping for tips that people have used personally and more down to earth explanations.

It's same as logaritm of a number with any other base. Assuming the result (logarithms value, the number it represents) is natural number, it describes how many "bases" do you have to "multiply by each other" in order to get the number of which logaritm we are taling about.

log (base) (some number) = 3 is another way of saying that (base) x (base) x (base) = (base)^3 = (some number).

It approximately tells you how many digits will number have if you write it down with given base.

As was surely stated before, natural logaritm (log. with base of e) of some set number is a constant - you don't have to count it, that already is pretty good description of the number.

Just like e(ulers number), it can be counted using infinite series - if you add up first few numbers from the series, you'll get decent approximation.

The series are somewhat easy to remember: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! .....
(5! = 5x4x3x2x1 and so on... beware! 0! = 1 - convention)

So first five numbers added up give you 1/1 + 1/1 + 1/2 + 1/6 + 1/24 = 2.708333... which is already pretty close to 2.71...

It is mostly important in derivations - in fact that is probably the main reason why is it used - e^x is one of very few functions (possibly only one - not sure now what happens when you multiply it by -1 and so) with an interesting property - its "elevation" / derivation, if you will, equals its value for every given x - in other words the speed at which it is increasing is (in certain sense) equal to it's value.

Natural logaritm is simply used to alter the exponent (the inside function) in order to have better looking outside function (e^something instead of (other base)^(something else)) so that you can use the unique properties of e^x...

Uh, ask on.... probably won't help much.... but worth a try

[noodle0117]

Here is a seemingly simplistic problem which I found myself stumbling over after a few minutes.

Find the integral of cos²x/sin³x

Someone in my class said they found the answer through substitution, but I am utterly unable to make any progress towards finding an answer for this integral.

[Tarqiup Inua]

Here is a seemingly simplistic problem which I found myself stumbling over after a few minutes.

Find the integral of cos²x/sin³x

Someone in my class said they found the answer through substitution, but I am utterly unable to make any progress towards finding an answer for this integral.

Perhaps using cos²x=1-sin²x might help?

I'd try substitution of x=cos x/ sin x, its derivation seems ugly only until you realize that using aformentioned equality gives you numerator of -1. Then maxbe per partes? Does it work?

EDIT: Well, the more I think about this the more I think this probably won't work..

[Karlito]

Perhaps using cos²x=1-sin²x might help?

I think this is a good route to pursue. It lets you split the problem into two different integrals. csc³ can be solved with integration by parts (treat it as the product of csc and csc²), and you probably already have the integral of csc in a table.

There might be a more elegant way, but alas, it's been a while since I had to actually evaluate integrals.