How do you simplify the integral after trigonometric substitution?

[nweis84]

I’ve got this tricky trig substitution that I’ve been trying to do as for my Calc II class. I think that I’ve done the substitution part right I just have no idea where to go from here. I’ve tried many routes including integrating by parts as well.

Please help me thank you

1.) Original question = ∫e^x * √(1+e^2x ) dx



2.) after trig substitution = ∫tanθ *[sec〗^3 θdθ

I have looked at another source for help with I believe an identical question and it integrates this and it gives me

3.) (〖sec〗^3 θ)/3 + C

So I guess all I’m really asking is how they got from step two to step three.

 

[Mark44]

Did you try the ordinary substitution u = e^x?
This gives du = e^x dx, so the integral becomes
$\int \sqrt{1 + u^2} du$

This integral can be evaluated using a trig substitution.

 

[nweis84]

no I've not tried that one yet thank you!

 

[Mark44]

Always try the simplest techniques first before tackling an integral with the more complicated techniques like trig substitution, or integration by parts, or partial fractions. If an ordinary substitution doesn't work, at least you haven't wasted much time.