Calculating Integrals with Tricky Substitutions

happyg1
Messages
304
Reaction score
0
hi,
I'm working on thiis:
.25 = \int_{-\infty}^m\frac{e^{x}}{(1+e^{-x})^2}dx
I let u=1+e^{-x} , du=-e{-x}dx
which gives:
\frac{1}{u}=\frac{1}{1+e^{-x}}|_{-\infty}^m
then I solve it out and I get
\frac{1}{1+e^{-m}}-1=.25
I have tried to solve this and I keep getting a negative ln. I moved the 1 to the RHS and then took ln of both sides. I can't get it to work.
help
CC
 
Physics news on Phys.org
But you don't have an e-x in the numerator. Try multiplying the top and bottom by e2x, then using the substitution u=1+ex, which will leave you with something like (u-1) to some power divided by u2.

By the way, there is no solution to the equation you mention at the end since 1/(1+ex)<1 for any real x.
 
ok
Statusx, you are absolutely correct. I have forgotten my - sign. It should read

.25 = \int_{-\infty}^m\frac{e^{-x}}{(1+e^{-x})^2}dx
sorry...
then let

u=1+e^{-x} , du=-e^{-x}dx
which gives
\frac{1}{1+e^{-m}}-1=.25
so then you get
\frac{1}{1+e^{-m}}=1.25
from there I took to ln of both sides and tried to solve for m, but nothing is working out.
CC
 
then,
1=1.25+1.25e^{-m}
then
-.2=e^{-m}
ln makes no sense
CC
 
It makes no sense because you've been sloppy!
We have:
\int_{-\infty}^{m}\frac{e^{-x}dx}{(1+e^{-x})^{2}}=\frac{1}{1+e^{-x}}\mid_{-\infty}^{m}=\frac{1}{1+e^{-m}}-\frac{1}{1+e^{\infty}}=\frac{1}{1+e^{-m}}
 
der duh der duh
\frac{1}{e^{-{-\infty}}}=0
I am a sloppy, sloppy student.:redface:
I'm all good now.
Thanks
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top