Question

Evaluate each of the following integrals. \int\frac{e^{x}}{1+e^{x}}dx

Applications of integrals
ANSWERED
asked 2021-05-23
Evaluate each of the following integrals.
\(\int\frac{e^{x}}{1+e^{x}}dx\)

Expert Answers (1)

2021-05-24
Step 1
To evaluate each of the following integrals.
Step 2
Given that
\(\int\frac{e^{x}}{1+e^{x}}dx\)
Let \(1+e^{x}=t\)
\(e^{x}=\frac{dt}{dx}\)
\(\Rightarrow dt=e^{x}dx\)
Hence we have
\(\int\frac{e^{x}}{1+e^{x}}dx=\int\frac{1}{t}dt\)
\(=\ln|t|+c\)
\(=\ln|1+e^{x}|+c\)
\(\therefore\int\frac{e^{x}}{1+e^{x}}dx=\ln|1=e^{x}|+c\)
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