Question

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

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

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
$$\int\frac{e^{x}}{1+e^{x}}dx=\int\frac{1}{t}dt$$
$$=\ln|t|+c$$
$$=\ln|1+e^{x}|+c$$
$$\therefore\int\frac{e^{x}}{1+e^{x}}dx=\ln|1=e^{x}|+c$$