Is There a Simpler Method for Integrating xe^0.1x?

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In summary, the given integral can be solved using either u-substitution or integration by parts. The latter method can be used to directly solve the integral without making a guess at the form of the answer.
  • #1
morson
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Homework Statement



[tex] \int {x}{e^{0.1x}} dx[/tex]

Homework Equations



U-substitution, differentiating.

The Attempt at a Solution



We have [tex] \int {x}{e^{0.1x}} dx[/tex]

Let [tex]u = 0.1x[/tex] therefore [tex]du = 0.1 dx[/tex] ==> [tex]dx = 10du[/tex]

Substituting back into the equation and using the fact that [tex]x = 10u[/tex]:

[tex] \int {10u}{e^{u}} 10 du[/tex] = [tex]100 \int {u}{e^u} du[/tex]

At this point I'm stuck. Is there another, simpler method?
 
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  • #2
Use integration by parts.

Let
u=x
dv=e^(0.1x) dx
 
  • #3
This looks OK.

To proceed further, what is another (besides substitution) technique of integration?
 
  • #4
George Jones said:
This looks OK.

To proceed further, what is another (besides substitution) technique of integration?

I haven't been studying integration for very long, but I learned a method where you differentiate one part of the integral and express the integral in terms of the derivative, and then use the fact that [tex]\int f'(x) = f(x) + C[/tex]

I'll have a shot:

From [tex] \int {x}{e^{0.1x}} dx[/tex]

[tex]\frac {d}{dx} {x}{e^{0.1x}} = {e^{0.1x}} + {0.1}{x}{e^{0.1x}} = {e^{0.1x}}({1} + {0.1}{x})[/tex]

I don't know how to express the integral in terms of f'(x), though.
 
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  • #5
You have found
[tex]\frac {d}{dx} xe^{0.1x} = e^{0.1x}(1 + 0.1x)[/tex]

Multiply that by 10 and integrate both sides:

[tex]10x{e^{0.1x}} = \int 10 e^{0.1x}dx + \int xe^{0.1x} dx[/tex]

You know how to integrate

[tex]\int 10 e^{0.1x}dx[/tex]

This integration technique amounts to making a guess at what type of function the answer will be, and (if you guess right) reducing the problem to a simpler one.

For this problem the standard method of integration by parts (which you might not have learned yet) will produce the answer without the need to guess what form it might take.
 
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1. How do I integrate this function?

Integrating a function involves finding the antiderivative or indefinite integral of the function. This can be done using various methods such as substitution, integration by parts, or using special integration formulas.

2. What is the purpose of integration in science?

Integration is used in science to determine the total amount or accumulation of a quantity over a given interval. It is also used to find the average value of a function and to solve various mathematical and scientific problems.

3. Can I use a calculator to integrate?

Yes, there are many calculators and software programs available that can accurately integrate functions. However, it is important to understand the concepts and methods behind integration in order to use these tools effectively.

4. What is the difference between definite and indefinite integration?

Definite integration involves finding the area under a curve between two specific points, while indefinite integration involves finding the antiderivative of a function without any specific limits. In other words, definite integration gives a numerical value, while indefinite integration gives a function as the solution.

5. How do I know which integration method to use?

The choice of integration method depends on the complexity of the function and the available tools. It is important to understand the different methods and their applications in order to determine the most efficient and accurate approach for a given function.

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