Integral of e^(3x)*arctan(e^(x))dx

[pamw]

Hi,

[integral]e^(3x)*arctan(e^(x))dx[/integral]

Should I turn e^(3x) into (e^(x)^3)? Should I substitute? Help please!

 

[Unco]

Actually, e^(3x) = (e^(x))^3. But making a substitution using this is indeed a good place to start. Have a go at it yourself and show us how you go.

 

[skeeter]

let \(\displaystyle \L t = e^x\)

\(\displaystyle \L dt = e^x dx\)

\(\displaystyle \L \int e^{3x} \arctan{(e^x)} dx =\)

\(\displaystyle \L \int e^{2x} \arctan{(e^x)} e^x dx\)

substitute ...

\(\displaystyle \L \int t^2 \arctan{(t)} dt\)

now use integration by parts ...

\(\displaystyle \L u = \arctan{(t)}\) and \(\displaystyle \L dv = t^2 dt\)