Khadija Wells

## Answered question

2021-10-13

Evaluate the indefinite integral.
$\frac{{e}^{x}dx}{{\left({e}^{x}+1\right)}^{4}}$

### Answer & Explanation

Jozlyn

Skilled2021-10-14Added 85 answers

Step 1
To evaluate:
$\frac{{e}^{x}dx}{{\left({e}^{x}+1\right)}^{4}}$
Step 2
Solving by applying u substitution:
Let $u={e}^{x}+1$
$⇒du={e}^{x}dx$
Hence we get:
$\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=\int \frac{1}{{\left(u\right)}^{4}}du$
Step 3
Solving the above integral, we get:
$⇒\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=\int {u}^{-4}du$
$⇒\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=\frac{{u}^{-4+1}}{-4+1}+C$
$⇒\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=\frac{{u}^{-3}}{-3}+C$
Step 4
Plugging the value of u, we get:
$⇒\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=\frac{{\left({e}^{x}+1\right)}^{-3}}{-3}+C$
$⇒\int \frac{{e}^{x}}{{\left({e}^{x}+1\right)}^{4}}dx=-\frac{1}{3{\left({e}^{x}+1\right)}^{3}}+C$
Step 5
Final Answer:
$\int \frac{{e}^{x}}{{\left({e}^{x+1}\right)}^{4}}dx=-\frac{1}{3{\left({e}^{x}+1\right)}^{3}}+C$

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