What is the Integral of 9/(3e^-6x + 3e^6x)?

[Suitengu]

Homework Statement

integrate 9/(3e^-6x + 3e^6x)

The Attempt at a Solution

integral :- the integral sign

integral [ (9/3)(1/e^6x + e^6x)]

3*integral [ 1/((1+e^12x)/e^6x)]

3*integral [ (e^6x)/(1+e^12x)]

Im stuck there. I thought I would get to something which I could integrate but I am at a loss.

 

[dynamicsolo]

3*integral [ (e^6x)/(1+e^12x)] dx

Im stuck there. I thought I would get to something which I could integrate but I am at a loss.

It looks like it's time for a u-substitution! What could you use for u that would leave you with an integral you'd know how to do?

 

[Suitengu]

I don't think that anything works as I did see that route as well. As a matter of fact the only thing i saw was u = 1 +e^12x but that does work as the derivative of that is 12e^12x which is not in the numerator and from what I see, I doubt I can arrange the numerator to look like that.

 

[dynamicsolo]

I don't think that anything works as I did see that route as well. As a matter of fact the only thing i saw was u = 1 +e^12x but that does work as the derivative of that is 12e^12x which is not in the numerator and from what I see, I doubt I can arrange the numerator to look like that.

In using u-substitutions, you don't always want to swallow the entire denominator in one bite. Try u = e^(6x) ...

 

[Suitengu]

O.O. whoops you are right. then that would be

u = e^6x
du = 6e^6x dx

3*(1/6)*integral [1/(1+u^2)] which is a inverse trig integral.

(1/2)*arctan(u)

(1/2)*arctan(e^6x)

aright thanks man. How do I signal that I have gotten help and found a solution again?

 

[dynamicsolo]

O.O. whoops you are right. then that would be

u = e^6x
du = 6e^6x dx

3*(1/6)*integral [1/(1+u^2)] which is a inverse trig integral.

(1/2)*arctan(u)

(1/2)*arctan(e^6x)

aright thanks man. How do I signal that I have gotten help and found a solution again?

...and don't forget the +C . ;-) (I always mention this because I've seen students get dinged on homework and exams for leaving that off of indefinite integral results.)

As the OP, you can edit the title of your thread. The custom here is to add '[SOLVED]' at the beginning of the title.

The u-substitution method is pretty versatile and can be applied in sometimes surprising situations. It can be a good idea to practice as many types of integrals in that section of your book/course as you can spare time for, in order to get experience with some of the kinds of substitutions you should be alert for.

 

[rocomath]

nice problem, ima borrow it if you don't mind ;)

 

[Suitengu]

yeah don't worry I am a quite apt person when it comes to these things and it seems i had just forgotten how versatile as you said it is. and as for the C, i was lacerated badly by a teacher for leaving it off and haven't done it since so i am good. thanks to one and all again.