Question

$$\int \frac{e^x}{{(1+e^x)}^2}dx$$

Answer 1

$$\int \frac{e^{x}dx}{{(1+e^x)}^2}$$

$$\text{Let 1 }+e^x=t$$

$$\Rightarrow e^x=\frac{dt}{dx}$$

$$\Rightarrow e^{x}dx=dt$$

$$Now,\int \frac{e^{x}dx}{{(1+e^x)}^2}$$

$$=\int \frac{dt}{t^2}$$

$$=\int t^{-2}dt$$

$$=\frac{t^{-2}+1}{-2+1}+C$$

$$=\frac{-1}{t}+C$$

$$=-\frac{1}{1+e^x}+C$$