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

\(\int\frac{e^{x}}{1+e^{x}}dx=\int\frac{1}{t}dt\)

\(=\ln|t|+c\)

\(=\ln|1+e^{x}|+c\)

\(\therefore\int\frac{e^{x}}{1+e^{x}}dx=\ln|1=e^{x}|+c\)

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

\(\int\frac{e^{x}}{1+e^{x}}dx=\int\frac{1}{t}dt\)

\(=\ln|t|+c\)

\(=\ln|1+e^{x}|+c\)

\(\therefore\int\frac{e^{x}}{1+e^{x}}dx=\ln|1=e^{x}|+c\)