# In the given equation as follows , use a table

hunterofdeath63 2021-12-10 Answered
In the given equation as follows , use a table of integrals with forms involving the trigonometric functions to find the indefinite integral:
$\int \frac{1}{1+{e}^{2x}}dx$
You can still ask an expert for help

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it

## Expert Answer

Bertha Jordan
Answered 2021-12-11 Author has 37 answers

Step 1
Let $1+{e}^{2x}=t$. Differentiate both sides.
${e}^{2x}×2dx=dt$
$dx=\frac{dt}{2{e}^{2x}}$
$=\frac{dt}{2\left(t-1\right)}$
Substitute the values into the integral. Also since the integral is indefinite; a constant of integration is to be added.
Step 2
Perform the integration.
$\int \frac{1}{1+{e}^{2x}}dx=\int \frac{1}{t}×\frac{dt}{2\left(t-1\right)}$
$=\frac{1}{2}\int \frac{dt}{t\left(t-1\right)}$
$=\frac{1}{2}\int \left[\frac{1}{t-1}-\frac{1}{t}\right]dt$
$=\frac{1}{2}×\left[\mathrm{ln}|t-1|-\mathrm{ln}|t|\right]+c$
$=\frac{1}{2}×\left[\mathrm{ln}|1+{e}^{2x}-1|-\mathrm{ln}|1+{e}^{2x}|\right]+c$
$=\frac{1}{2}×\left[\mathrm{ln}|{e}^{2x}|-\mathrm{ln}|1+{e}^{2x}|\right]+c$
Hence the solution is obtained.

###### Not exactly what you’re looking for?
SlabydouluS62
Answered 2021-12-12 Author has 52 answers
$\int \frac{1}{{e}^{2x}+1}dx$
$=-\frac{1}{2}\int \frac{1}{{e}^{-u}+1}du$
$=\int \frac{{e}^{u}}{{e}^{u}+1}du$
$=\int \frac{1}{v}dv$
$=\mathrm{ln}\left(v\right)$
$=\mathrm{ln}\left({e}^{u}+1\right)$
$-\frac{1}{2}\int \frac{1}{{e}^{-u}+1}du$
$=-\frac{\mathrm{ln}\left({e}^{u}+1\right)}{2}$
Answer:
$=-\frac{\mathrm{ln}\left({e}^{u}+1\right)}{2}+C$
###### Not exactly what you’re looking for?

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it