Introduction to Calculus Tutorial - Page 4

^_^Engineer_Adam^_^ · #31 $Old$ 05-28-2007, 06:36 AM

Quote:

Originally Posted by ThePerfectHacker $View Post$

Find the integrals.

4*) $\int e^{\sqrt{x}}$ It has a star what do you think!?

10) $\int \frac{1}{1-e^{2x}}dx$ Hard

hi
i cant seem to solv this two problems can u please provide solution to these?

arrgh

ThePerfectHacker · #32 $Old$ 05-28-2007, 09:32 AM

Quote:

Originally Posted by ^_^Engineer_Adam^_^ $View Post$

hi
i cant seem to solv this two problems can u please provide solution to these?

arrgh

$\int e^{\sqrt{x}} dx$
I will make the substitution $u=\sqrt{x}$ . But if I do that, I always need to know what its derivative is, $u' = \frac{1}{2\sqrt{x}}$ .

If you look into the integral this factor does not appear. So what do we do? We make it appear. Multiply the numerator and denominator by this expression:

$\int e^{\sqrt{x}}\cdot 2\sqrt{x} \cdot \frac{1}{2\sqrt{x}} dx$
Now if we use $u=\sqrt{x}$ then, $u'=\frac{1}{\sqrt{x}}$ which is good because it appears as a factor and $2\sqrt{x} = 2u$ .
Make the substitution,
$\int e^u \cdot 2u \cdot u' dx = \int 2u e^u du$ .
You can now do this integral by parts.

^_^Engineer_Adam^_^ · #33 $Old$ 05-30-2007, 04:47 PM

i get it now
$\int \frac{1}{1-e^{2x}}dx$

$\int \frac{1 - e^{2x}+e^{2x}}{1-e^{2x}}dx$

$\int \frac{1-e^{2x}}{1-e^{2x}}dx + \frac{e^{2x}}{1-e^{2x}}dx$

$x + \int \frac{e^{2x}}{u} \frac{du}{-2e^{2x}}$

$x - \frac{1}{2}\ln(1-e^{2x}) +C$

but the integrator says

$x - \frac{1}{2}\ln(e^{2x}-1)$
hmm

ThePerfectHacker · #34 $Old$ 05-30-2007, 04:58 PM

Quote:

Originally Posted by ^_^Engineer_Adam^_^ $View Post$

i get it now
$\int \frac{1}{1-e^{2x}}dx$

$\int \frac{1 - e^{2x}+e^{2x}}{1-e^{2x}}dx$

$\int \frac{1-e^{2x}}{1-e^{2x}}dx + \frac{e^{2x}}{1-e^{2x}}dx$

$x + \int \frac{e^{2x}}{u} \frac{du}{-2e^{2x}}$

$x - \frac{1}{2}\ln(1-e^{2x}) +C$

but the integrator says

$x - \frac{1}{2}\ln(e^{2x}-1)$
hmm

Because you need to use $| \, |$ .
In that case,
$\ln |1-e^{2x}| = \ln |e^{2x}-1|$
Because,
$|a-b|=|b-a|$

tukeywilliams · #35 $Old$ 06-30-2007, 01:06 AM

just wanted to point out: isn't a sequence a function defined as $f: \mathbb{Z^{+}} \rightarrow A$ which means a sequence in the set $A$ ? So the codomain doesn't have to be $\mathbb{R}$ but can be an arbitrary set $A$ ? Actually $\mathbb{N}$ and $\mathbb{Z^{+}}$ are equivalent right? And a sequence is null when $\lim_{n \rightarrow \infty} f(n) = 0$ when $\forall \epsilon \in \mathbb{R^{+}}, \exists N \in \mathbb{Z^{+}}, \forall n \in \mathbb{Z^{+}} (n \geq N \Rightarrow |f(n)| < \epsilon)$ . Also, I like to think of functions as follows: pretend you have a box subdivided into smaller boxes. These smaller boxes represent the elements of the codomain, and the points in the boxes represent the elements of the domain. So a function orders the points in the smaller boxes.

ThePerfectHacker · #36 $Old$ 06-30-2007, 07:58 PM

Quote:

Originally Posted by tukeywilliams $View Post$

just wanted to point out: isn't a sequence a function defined as $f: \mathbb{Z^{+}} \rightarrow A$ which means a sequence in the set $A$ ? So the codomain doesn't have to be $\mathbb{R}$ but can be an arbitrary set $A$ ? Actually $\mathbb{N}$ and $\mathbb{Z^{+}}$ are equivalent right? And a sequence is null when $\lim_{n \rightarrow \infty} f(n) = 0$ when $\forall \epsilon \in \mathbb{R^{+}}, \exists N \in \mathbb{Z^{+}}, \forall n \in \mathbb{Z^{+}} (n \geq N \Rightarrow |f(n)| < \epsilon)$ . Also, I like to think of functions as follows: pretend you have a box subdivided into smaller boxes. These smaller boxes represent the elements of the codomain, and the points in the boxes represent the elements of the domain. So a function orders the points in the smaller boxes.

I was talking about a "real sequence" in that case $f:\mathbb{N}\to \mathbb{R}$ .

colby2152 · #37 $Old$ 01-04-2008, 01:45 PM

TPH, your notes are extremely thorough and detailed, and are an excellent source for anyone studying Calculus. I hope you don't mind that I added some of my "simpler" Calculus notes to this forum. Like yours, my notes are not meant for students taking Advanced Calculus courses such as Real Analysis. I hope I am not stepping on your toes!

Jhevon · #38 $Old$ 01-04-2008, 01:49 PM

Quote:

Originally Posted by colby2152 $View Post$

TPH, your notes are extremely thorough and detailed, and are an excellent source for anyone studying Calculus. I hope you don't mind that I added some of my "simpler" Calculus notes to this forum. Like yours, my notes are not meant for students taking Advanced Calculus courses such as Real Analysis. I hope I am not stepping on your toes!

that would be all good and well i believe. in fact, we encourage users to do things like that. that's why we have a mathwiki page (but it's not fully functional yet). as long as you're not "re-inventing the wheel" (that is, going over exactly the same stuff TPH covered, unless of course, you feel he left something out) i don't see why you can't post notes. you may want to do it in a different thread though, this is TPH's thread.

mustkara · #39 $Old$ 02-07-2008, 04:13 AM

I believe the textbook

Introduction to Stochastic Calculus with Applications, 1st Edition

will be useful for you to study Mathematics

Good Luck and wish help for you.

hehe ^_^