Integrating xe^{ax}: A Step-by-Step Solution

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In summary, the conversation is about finding the integral of xe^{ax}. The homework equations used are integration by substitution and integration by parts. The person asking for help shows their attempt at a solution and the correct answer is given. Mistakes are pointed out and the person asking for help corrects them and thanks for the assistance. The final answer is \frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2} and the mistake made was with the derivative.
  • #1
Radarithm
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Homework Statement



Evaluate: [tex]\int{xe^{ax}}dx[/tex]

Homework Equations



Integration by substitution

The Attempt at a Solution



I'm on a phone at the moment. My work: http://postimg.org/image/v4hdr5uqx/

The correct answer was:
[tex]\frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}[/tex]
 
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  • #2
You should do this integral by parts. Some of your original substitutions don't look OK.
 
  • #3
http://postimg.org/image/ao3mi4ygz/

I feel like I'm getting closer but I'm still making a dumb mistake. Is it with the derivatives?
 
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  • #4
If [itex]dv = e^{ax} dx[/itex], then [itex]v = e^{ax}[/itex] is wrong. There are also multiple errors on the second line, but you need to fix what I said first.
 
  • #5
Got it. Thanks for pointing out my mistakes.
[tex]\int{xe^{ax}}dx[/tex] [tex] u=x du=dx dv=e^{ax}dx v=\frac{e^{ax}}{a} [/tex]
[tex]\frac{xe^{ax}}{a}-\int{\frac{e^{ax}}{a}}dx[/tex]
[tex]\frac{xe^{ax}}{a}-\int{e^{ax}a^{-1}}dx=\frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}[/tex]

Power rule + derivative mistake
Thanks for the help.
 

FAQ: Integrating xe^{ax}: A Step-by-Step Solution

1. What is the purpose of integrating xe^{ax}?

The purpose of integrating xe^{ax} is to find the antiderivative or the original function from its derivative. Integration is the inverse operation of differentiation and it allows us to solve problems involving rates of change, such as finding the displacement, velocity, or acceleration of an object.

2. What is the step-by-step process for integrating xe^{ax}?

The step-by-step process for integrating xe^{ax} involves first applying the power rule of integration to the exponential function, which results in e^{ax}/a. Then, we use the product rule to solve for the integral of xe^{ax} by splitting it into two parts: x times e^{ax} and e^{ax} times dx. Finally, we use integration by parts to solve for the integral of x times e^{ax} and substitute the results back into the original equation to find the final solution.

3. What is the general formula for integrating xe^{ax}?

The general formula for integrating xe^{ax} is ∫xe^{ax} dx = xe^{ax}/a - ∫e^{ax} dx.

4. Are there any special cases for integrating xe^{ax}?

Yes, there are two special cases for integrating xe^{ax}: when a = 0 and when a = 1. When a = 0, the integral becomes ∫xe^{0} dx = x + C, where C is the constant of integration. When a = 1, the integral becomes ∫xe^{x} dx = xe^{x} - e^{x} + C.

5. How is integrating xe^{ax} used in real life applications?

Integrating xe^{ax} is used in various fields of science and engineering, such as physics, economics, and finance. For example, in physics, it can be used to calculate the displacement and velocity of a moving object, while in economics, it can be used to solve problems involving compound interest. It is also used in signal processing to analyze and manipulate signals in communication systems.

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