Given the indefinite integral:
[tex]\int\frac{5e^{5x}}{1+e^{10x}}dx[/tex]
We use the following change of variables:
[tex]\begin{gathered} u=e^{5x} \\ \Rightarrow du=5e^{5x}dx \end{gathered}[/tex]
Then:
[tex]\int\frac{5e^{5x}}{1+e^{10x}}dx=\int\frac{5e^{5x}dx}{1+e^{10x}}=\int\frac{du}{1+u^2}[/tex]
This is a known integral:
[tex]\int\frac{du}{1+u^2}=\arctg(u)+C[/tex]
Finally, going back to our original variable:
[tex]\int\frac{5e^{5x}}{1+e^{10x}}dx=\arctg(e^{5x})+C[/tex]
