Need some help with Calculus

by Rishi Sun Mar 31, 2013 2:47 pm

What is the anti derivative for

-(e ** x / e ** (x**2)) dx

Thanks

by MaxEntropy_Man Sun Mar 31, 2013 2:54 pm

if you want the indefinite integral, there is no closed form analytical solution. it is in terms of the error function which itself is defined as an integral. the exact result from mathematica is:

-(1/2) E^(1/4) Sqrt[\[Pi]] Erf[1/2 (-1 + 2 x)]

and here is the definition of the error function:

http://en.wikipedia.org/wiki/Error_function

by MaxEntropy_Man Sun Mar 31, 2013 3:14 pm

you might find this useful to check answers. i don't recommend it as a learning tool:

http://integrals.wolfram.com/index.jsp?expr=-Exp%5Bx%5D%2FExp%5Bx%5E2%5D&random=false

by MaxEntropy_Man Sun Mar 31, 2013 10:52 pm

are you helping a kid with calc? if so is it for an AP calc class? were my answers of any use?

by Rishi Sun Mar 31, 2013 11:23 pm

I am following the MOOC course Calculus I on Coursera.org.

The problem was an exercise in using reverse chain rule to find the anti-derivative.

Also apply the principle of composition of two functions f(x) and g(x)

Problem: Find the anti-derivative of -((e^x)/(e^x^2)) dx

The derivative of the function of the form f(g(x))

d/dx f(g(x)) = f'(g(x))*g'(x)

if we let g'(x) = e^x and f'(g(x)) = -(e^x^2)

Then f(x) = 1/x and g(x)=e^x

Therefore anti-derivative of -((e^x)/(e^x^2)) dx = 1/(e^x)

by Uppili Mon Apr 01, 2013 12:30 am

Rishi wrote:What is the anti derivative for

-(e ** x / e ** (x**2)) dx

Thanks

Not that I remember, but substituting y = e**x should result in - (dy)/y^2 and you should be able to integrate fairly easily.

P.S. read my signature.

by Kris Mon Apr 01, 2013 1:02 am

Uppili wrote:
Rishi wrote:What is the anti derivative for

-(e ** x / e ** (x**2)) dx

Thanks

Not that I remember, but substituting y = e**x should result in - (dy)/y^2 and you should be able to integrate fairly easily.

P.S. read my signature.

>>> I don't remember any of this and I was not too shabby at it either. Oh well..

by Uppili Mon Apr 01, 2013 1:09 am

Kris wrote:
Uppili wrote:
Rishi wrote:What is the anti derivative for

-(e ** x / e ** (x**2)) dx

Thanks

Not that I remember, but substituting y = e**x should result in - (dy)/y^2 and you should be able to integrate fairly easily.

P.S. read my signature.

>>> I don't remember any of this and I was not too shabby at it either. Oh well..

There was a time I loved integration and Series, and I could do any problem sleeping. Now, if I even look at any problem I fall asleep.

by MaxEntropy_Man Mon Apr 01, 2013 6:27 am

i misinterpreted your problem because you wrote e**(x**2). that is not the same as (e^x)^2 which is what the problem seems to have been. so in essence the problem is to find the integral of -e^(-x). which is e^(-x).

by Maria S Wed Apr 03, 2013 3:11 pm

I came across the link to this website..it's nice..seems to have a lot of interesting stuff (you may all know about this!)

http://www.openculture.com/

*I have no idea, if the math courses are helpful or not! Thought I'll post the link here.

http://www.openculture.com/math_free_courses

by Seva Lamberdar Wed Apr 03, 2013 3:39 pm

Rishi wrote:I am following the MOOC course Calculus I on Coursera.org.

The problem was an exercise in using reverse chain rule to find the anti-derivative.

Also apply the principle of composition of two functions f(x) and g(x)

Problem: Find the anti-derivative of -((e^x)/(e^x^2)) dx

The derivative of the function of the form f(g(x))

d/dx f(g(x)) = f'(g(x))*g'(x)

if we let g'(x) = e^x and f'(g(x)) = -(e^x^2)

Then f(x) = 1/x and g(x)=e^x

Therefore anti-derivative of -((e^x)/(e^x^2)) dx = 1/(e^x)

Better check the answer again.

If you differentiate the expression on the right hand side (1/(e^x)), you don't get what you wrote on the left hand side.

by Hellsangel Wed Apr 03, 2013 3:53 pm

Sevaji, the suspense is killing me. What is the right answer according to you?

by Seva Lamberdar Wed Apr 03, 2013 4:09 pm

HAji, "enjoy" the suspense while waiting for Rishiji to clarify the things.

by Hellsangel Wed Apr 03, 2013 4:11 pm

Seva Lamberdar wrote:HAji, "enjoy" the suspense while waiting for Rishiji to clarify the things.

Sevaji, using racial slurs now?

PS:I am not even Iraqi.

PPS: The solution is right.

by Rishi Wed Apr 03, 2013 5:16 pm

Sevaji

I will recheck and confirm.

by Seva Lamberdar Wed Apr 03, 2013 5:33 pm

Rishi wrote:Sevaji

I think there is a typo in the problem itself.

The website had the problem listed as

-((e^x)/(e^x2)) dx

Look at the denominator carefully. It was originally posted as e^x2.

I thought it should be e^x **2

But it looks like you are correct. It should be e^2x. Instead of the power being 2x, the instructor posted it as 2x.

The problem was given this way to able to decompose and use reverse chain tule.

Of course Rishi ji, the denominator in the above should read as (e^x)^2 or e^2x (and not e^x^2) to arrive the solution as 1/(e^x).

Btw we can also find the "exact" solution (integrand or anti-derivative) for the problem in the above form, i.e. Int [(e^x)/(e^x^2)]dx, if x is small (x<<1) such that the cubic and higher order terms in the expansions of e^x and e^x^2 can be neglected.

Then, for small x,

Int (e^x)/(e^x^2)]dx = arctan x +/- (1/2) ln (1 + x^2) + (1/2) {+/- x -/+ arctan x}

by Rishi Wed Apr 03, 2013 6:34 pm

Correction

Instead of the power being 2x, the instructor posted it as 2x.

Should be

Instead of the power being 2x, the instructor posted it as x2.

by Rishi Wed Apr 03, 2013 6:35 pm

Sevaji,

Thanks for clarification.

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