Integrating $\int\frac{xe^{2x}}{(2x+1)^2}dx$

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In summary, the problem requires finding the indefinite integral of xe^{2x}/(2x+1)^2, using substitution and integration by parts. The suggested substitution of u=xe^{2x} and du=2xe^{2x}+e^{2x}dx does not work, but using integration by parts with u=x*exp(2x) and dv=dx/(2x+1)^2 yields the correct solution.
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Homework Statement


[tex]\int\frac{xe^{2x}}{(2x+1)^2}dx[/tex] where "e" is the natural number


Homework Equations


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The Attempt at a Solution


I tried many ways to solve this problem, but to no avail.
the hint on the book said to use substitution and make [tex]u=xe^{2x}[/tex] and [tex]du=2xe^{2x}+e^{2x}dx[/tex] but I don't see how that would work out; there is no way to change all the x's into u's.
 
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I don't think they meant a simple substitution. They meant to integrate by parts with u=x*exp(2x). Pick dv=dx/(2x+1)^2. That works.
 

FAQ: Integrating $\int\frac{xe^{2x}}{(2x+1)^2}dx$

What is the general approach for integrating this type of function?

The general approach for integrating this type of function is to use substitution. In this particular case, let u = 2x+1, and then solve for x in terms of u. This will allow you to rewrite the integral in terms of u, making it easier to integrate.

Can integration by parts be used for this integral?

Yes, integration by parts can also be used for this integral. However, it may require multiple iterations and can become quite lengthy, so substitution is often a simpler approach.

Is there a specific technique for handling the exponential term?

There is not a specific technique for handling the exponential term in this integral. It can be treated like any other term and should be incorporated into your substitution or integration by parts method.

Are there any special cases or exceptions for this type of integral?

One special case is when the denominator of the fraction can be factored into linear terms. In this case, you can use partial fraction decomposition to simplify the integral and make it easier to integrate.

Is there a way to check if the solution is correct?

Yes, you can always check your solution by taking the derivative of the antiderivative you found. This should result in the original integrand. You can also use online calculators or software to verify your answer.

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