Struggling with Integrating (xe^(2x))/(1+2x)^2? Get a Helpful Tip Here!

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In summary, the conversation is about integrating the function xe^(2x)/(1+2x)^2, with the initial attempts at using integration by parts and substitution. A suggestion is made to try a second substitution, but it is clarified that the first substitution should be sufficient. The final result is a revised integral with a missing du term, which is then fixed by simplifying the integrand and reinstating the du term.
  • #1
autodidude
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Homework Statement


Integrate [tex]\frac{xe^{2x}}{(1+2x)^2}[/tex] with respect to x

Didn't get anywhere with integration by parts or substitution using u=xe^(2x)
A push in the right direction would be much appreciated.
 
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  • #2
Try v = 1 + 2x.
 
  • #3
As a second substitution?
 
  • #4
No, just start with it.
 
  • #5
Ok, I now have the following:

[tex]\frac{1}{4} \int \frac{(u-1)e^{(u-1)}{u^2}[/tex]
 
  • #6
autodidude said:
Ok, I now have the following:

[tex]\frac{1}{4} \int \frac{(u-1)e^{(u-1)}{u^2}[/tex]

Allow me to fix that for you:

##\displaystyle \frac{1}{4} \int \frac{(u-1)e^{(u-1)}}{u^2} \ du##
 
Last edited:
  • #7
where is du?
 
  • #8
autodidude said:

Homework Statement


Integrate [tex]\frac{xe^{2x}}{(1+2x)^2}[/tex] with respect to x

Didn't get anywhere with integration by parts or substitution using u=xe^(2x)
A push in the right direction would be much appreciated.

Integrate by parts

∫uv'dx=uv-∫u'vdx,

using u=xe2x and v'=1/(1+2x)2.

ehild
 
  • #9
ehild said:
Integrate by parts

∫uv'dx=uv-∫u'vdx,

using u=xe2x and v'=1/(1+2x)2.

ehild

Parts requires u,v to be continuous.
 
  • #10
Now that we have reinstated du, observe that e^(u - 1) = (e^u)/e; the 1/e constant goes outside, and what's inside can be simplified into ((e^u)/u - (e^u)/u^2).
 

1. What is the formula for integrating (xe^(2x))/(1+2x)^2?

The formula for integrating (xe^(2x))/(1+2x)^2 is given by:
∫ (xe^(2x))/(1+2x)^2 dx = (e^(2x)(2x^2+4x+1))/(4(1+2x)) + C

2. How do you simplify the integrated form of (xe^(2x))/(1+2x)^2?

To simplify the integrated form of (xe^(2x))/(1+2x)^2, you can use partial fractions and break the expression into simpler fractions.

3. Can the integral of (xe^(2x))/(1+2x)^2 be solved using substitution?

Yes, you can solve the integral of (xe^(2x))/(1+2x)^2 using substitution. Let u = 1+2x, then du/dx = 2 and dx = du/2. By substituting these values, the integral can be simplified.

4. Is there a shortcut or trick to solving the integral of (xe^(2x))/(1+2x)^2?

There is no specific shortcut or trick to solving the integral of (xe^(2x))/(1+2x)^2. However, using different methods such as substitution, partial fractions, or integration by parts can make the process easier.

5. How can the integral of (xe^(2x))/(1+2x)^2 be used in real-life applications?

The integral of (xe^(2x))/(1+2x)^2 is used in various fields of science and engineering, such as physics, chemistry, and economics. It can be used to solve problems related to growth and decay, motion, and optimization.

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