find expression for int [e^(2x) / sqrt(1+e^(2x))] dx

[val1]

Hi.
Can someone help with a couple of questions that I have? I am revising for an exam.

Find an expression for the indefinite integral \(\displaystyle \L
\int {\frac{{e^{2x} }}{{\sqrt {1 + e^{2x} } }}dx}\)

Thank you. :?

 

[galactus]

You could start by letting \(\displaystyle \L\\u=e^{2x}, \;\ du=2e^{2x}dx, \;\ \frac{du}{2}=e^{2x}dx\)

Make the substitutions and you have:

\(\displaystyle \L\\\frac{1}{2}\int\frac{1}{\sqrt{1+u}}du\)

Now, can you finish?.

 

[morson]

\(\displaystyle u = 1 + e^{2x}\), \(\displaystyle du = 2e^{2x}dx\), \(\displaystyle \frac{du}{2}\ = e^{2x}dx\)

= \(\displaystyle \L\ \frac{1}{2}\ \int u^{-1/2} \ du\)

= \(\displaystyle \L\ sqrt{1 + e^{2x}} + C\)