Question

Hi there, I am having some issues solving this question for calc-12

Answer 1

Given the indefinite integral:

`\int(5e^(5x))/(1+e^(10x))dx`

We use the following change of variables:

`\begin{gathered} u=e^(5x) \\ \Rightarrow du=5e^(5x)dx \end{gathered}`

Then:

`\int(5e^(5x))/(1+e^(10x))dx=\int(5e^(5x)dx)/(1+e^(10x))=\int(du)/(1+u^2)`

This is a known integral:

`\int(du)/(1+u^2)=\arctg(u)+C`

Finally, going back to our original variable:

`\int(5e^(5x))/(1+e^(10x))dx=\arctg(e^(5x))+C`