Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$

Question

I am trying to solve this integral and I need your suggestions.
I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now.

$$\int^1_0 \frac{dx}{1+e^{2x}}$$ Thanks!

Multiply the numerator and the denominator by $e^{-2x}$. – Random Variable May 14 '13 at 10:26 — Random Variable, May 14 '13 at 10:26

score 5 · Accepted Answer · answered May 14 '13 at 10:27

5

With the change of variable $u=e^{x}$, you get $$ \int_{[0,1]}\frac{dx}{1+e^{2x}} = \int_{[1,e]}\frac{1}{u(1+u^2)}du $$

answered May 14 '13 at 10:27

Clement C.

67,323

Then expand in partial fraction, and it's ok. – Jean-Claude Arbaut May 14 '13 at 10:30
how its $u^2$ ? – Ofir Attia May 14 '13 at 11:03
What do you mean? ($e^{2x}=(e^{x})^2=u^2$, if that's what you ask) – Clement C. May 14 '13 at 12:10
@Clement C. after I get $\frac{1}{u(1+u^2)}du$ how I solve it with partial fraction? – Ofir Attia May 16 '13 at 07:46
By writing it as a sum of (easy to integrate) fractions, and integrating each of them independently. – Clement C. May 16 '13 at 11:53

lab bhattacharjee · Answer 2 · 2013-05-14T10:40:48.547

2

HINT:

Putting $e^{2x}=v\implies 2x=\ln v\implies 2dx=\frac{dv}v$

$$\int^1_0 \frac{dx}{1+e^{2x}}=\int_1^{ e^2}\frac1{2v(1+v)}dv=\frac12\cdot\int_1^{ e^2}\left(\frac1v-\frac1{v+1}\right)dv $$

edited May 14 '13 at 10:40

answered May 14 '13 at 10:31

lab bhattacharjee

274,582

score 1 · Answer 3 · answered May 14 '13 at 12:29

1

$\int^{1}_{0}\frac{dx}{1+e^{2x}}=$ $ \int^{1}_{0}\frac{e^{-2x}dx}{1+e^{-2x}}=$ $-\frac{1}{2}\int^{1}_{0}\frac{(1+e^{-2x})'dx}{1+e^{-2x}}=$ $=-\frac{1}{2}ln(1+e^{-2x})|^{1}_{0}= \frac{1}{2}ln\frac{2e^{2}}{1+e^{2}}$

answered May 14 '13 at 12:29

medicu

4,482

Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$

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