"Evaluate the indefinite integral. exdx(ex+1)4" was answered by our community

Khadija Wells

Answered question

2021-10-13

Evaluate the indefinite integral.

\frac{e^{x} d x}{{(e^{x} + 1)}^{4}}

Answer & Explanation

Jozlyn

Skilled2021-10-14Added 85 answers

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

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