Solving the Hard Indefinite Integral: e^(3x) * sqrt(1+e^(2x))

[kppc1407]

1. Homework Statement [/b]

\int e^{3x}\sqrt{1+e^{2x}dx

Homework Equations

Substitution
Parts of Integration

The Attempt at a Solution

Started off using U substitution setting e^x = to u. Then tried to use parts of integration. Now I am stuck.

 

[Anonymous217]

This is a very messy problem.
After u-substitution (let u = e^{x}), you should get
\int u^2\sqrt{1+u^2}du

Then I imagine you should try u-substitution using trig functions like tan and sec. It gets very convoluted very quickly.

 

[kppc1407]

Thats what I have done and it seems to continue to get larger. Just trying to make sure I was on the right track. Thanks for your help

 

[Count Iblis]

Instead of committing fully to one particular approach, you should do some exploratory computations to see what works best. That will often save you from persuing some tedious method if a very simple method is available. In this case you missed a partial integration step where you integrate the factor u sqrt(1+u^2) and thus have to evaluate the integral of (1+u^2)^(3/2), which suggests substituting u = sinh(t) leaving you having to integrate cosh^4(t), which is trivial.

 

[kppc1407]

Haven't learned U=sinh(t). Only using u-sub, Trig sub, and parts. I think that's what is making it so long and messy. If any other suggestions it would be much appreciated.

 

[Anonymous217]

This requires hyperbolic sin.

 

[Bohrok]

Haven't tried this out, just the first thing I thought of:
e^3x√(1 + e^2x) = e^x·e^2x√(1 + e^2x)

You can try integration by parts, integrating the right side with the substitution u = 1 + e^2x

Whatever you do, you won't have to go into hyperbolic trig functions, even if the regular trig functions make the integration a little messy.