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

In summary, the conversation discusses a challenging integration problem involving the expression \int e^{3x}\sqrt{1+e^{2x}dx. The initial attempt involves using u-substitution and parts of integration, but the problem becomes convoluted. Another approach involving trigonometric substitution is suggested, but the conversation ultimately concludes that a simpler method using hyperbolic sin is available. The conversation also mentions the possibility of using integration by parts.
  • #1
kppc1407
19
0
1. Homework Statement [/b]

[tex]\int e^{3x}\sqrt{1+e^{2x}[/tex]dx

Homework Equations



Substitution
Parts of Integration

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

Then I imagine you should try u-substitution using trig functions like tan and sec. It gets very convoluted very quickly.
 
  • #3
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
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
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.
 
  • #6
This requires hyperbolic sin.
 
  • #7
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.
 

1. What is a hard indefinite integral?

A hard indefinite integral is an integral that cannot be easily evaluated using basic integration techniques such as substitution or integration by parts. It often involves complex algebraic expressions or trigonometric functions.

2. How do you solve a hard indefinite integral?

There is no one set method for solving a hard indefinite integral. It often requires a combination of various integration techniques and may also involve using advanced mathematical concepts such as series or numerical methods to approximate the solution.

3. What are some common strategies for solving a hard indefinite integral?

Some common strategies for solving a hard indefinite integral include using trigonometric identities, simplifying the integrand, and applying integration by parts multiple times. It may also be helpful to rewrite the integral in a different form or use substitution to simplify the expression.

4. Are there any online tools or resources for solving hard indefinite integrals?

Yes, there are various online integral calculators and resources available that can help with solving hard indefinite integrals. However, it is important to have a good understanding of integration techniques and concepts in order to use these tools effectively.

5. Why are hard indefinite integrals important in science?

Hard indefinite integrals are important in science because they allow us to solve complex mathematical problems and model real-world phenomena. They are often used in physics, engineering, and other scientific fields to calculate quantities such as velocity, acceleration, and work.

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