4 ways to integrate 1/(e^x+1)

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In today's video we are going to take a look at four different ways to integrate 1/(e^x+1). We are going to use partial fraction decomposition, integral identities, substitution eveness/oddness of functions! Enjoy :3

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[Chiinox]

Wow, Ive never seen decomposition of a function into an odd and an even function, very interesting, great video. :)

[kennkong]

You should show that the second and fourth answers reduce to the first, simplest form of the answer.

[gillkhaas]

i did it in a different way similar to your second way:

(1) let u=e^x gives du=e^xdx but since e^x=u this gives us u=udu
(2) now we have the integral of 1/(u+1)dx, if we times that by (u/u) we can substitute the u in the denominator and the dx by du,
this gives us the integral of 1/(u(u+1))du
(3) rewriting this using partial fractions gives us the integral of (1/u)du minus the integral of 1/(u+1)du
(4) solving these two integrals gives us ln(u)-ln(u+1)+c
(5) finally we substitute u for e^x, giving us a final answer of x-ln(e^x+1)+c

[palaknigam]

Thanks! You didn't give just solutions but taught me how to think for solving more similar problems.

[MrRyanroberson]

personally the third way is the most natural. i always used to try that with complicated fractions when first learning division, because identities clicked so naturally.
what is 256928/257446, well just 1-(518/257446)=1-1/497, which is easily 496/497.
the same expanded for complex numbers and integrals. not at all exotic.

[XgamersXdimensions]

Method two is definitely the first method I thought of trying because it doesn’t involve any trickery, but just brute forcing the integral. Nice to see the other methods though!

[EsKaioS]

2 years too late but thanks papa <3

[ciro]

The third method is awesome, a few steps and you can get around many such integrals🎉

[jensberling]

Kudos! Yes, the forth way is very charming and interesting. Thank you so much. You have my full support.

[apta]

Interestingly enough, the integral from 0 to infinity of that integrand is the integral that represents (Σ[n=1, Inf] (-1)^n * n^-s) * Γ(s) when s = 1

[Se_bito]

When I tried this integral the first thing I did was try the third method you used, algebraic manipulation is so nice in some cases.

[mrchejov]

Thanks you i send my warmest regards from Chile :3

[mokouf]

I really love 1st and 3rd methods for simplicity and not requiring partial fraction.
4th method...just weird, and I won't use that method, but it is valid though.

[seoexperimentations]

watching this after crushing my Calculus final cheers

[MrRyanroberson]

the way you did the fourth one reminds me strongly of the complex definition of sin(a+bi) and cosine

[saran]

Simply add and subtract e^x in the numerator, and then use substitution, bam!

[cpotisch]

Alternately, 1/(e^x + 1) = e^-x/(e^-x + 1) => u equals the denominator, -du = e^-xdx. Thus we get the integral of du/u, which is ln|e^-x + 1|

[jayantimajumder]

You should do some adjustments in rational integration.
1/(u)(u-1) is written as (u)-(u-1)/(u).(u-1). That is very easy and also a time savior. After all love your comedy 😂😂😂 .Your odd even in indefinite and definit integration definitely awesome 👍🙏

[ovnii_]

great vid man, i used the first one, this seemed more natural to me :)

[dome_dera]

thank you very much! when my teacher explained this i didn´t understant anything so i search the solution on youtube and i only understood this with you video uwu