Evaluate the indefinite integral. exdx(ex+1)4

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