Homework Help: Hard Indefinite Integral

1. Feb 20, 2010

kppc1407

1. The problem statement, all variables and given/known data[/b]

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

2. Relevant equations

Substitution
Parts of Integration

3. The attempt at a solution

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

2. Feb 20, 2010

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.

3. Feb 20, 2010

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

4. Feb 20, 2010

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.

5. Feb 21, 2010

kppc1407

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

6. Feb 21, 2010

Anonymous217

This requires hyperbolic sin.

7. Feb 22, 2010

Bohrok

Haven't tried this out, just the first thing I thought of:
e3x√(1 + e2x) = ex·e2x√(1 + e2x)

You can try integration by parts, integrating the right side with the substitution u = 1 + e2x

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.