Is using u-substitution the right approach for integrating e(x^2 + x)(2x+1) dx?

In summary, the conversation is about finding the integral of e(x^2 +x)(2x+1) dx and using the substitution method. The first attempt involved setting u = e(x^2 +x) and finding du = e(x^2 +x)(2x+1)dx, but the next step of using 1/u du did not match the professor's answer. The correct approach is to use u = e^{x^2+x} and du = e^{x^2+x}(2x+1) dx, which is equivalent to the original integral.
  • #1
a.a
127
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Homework Statement



integrate: e(x^2 +x)(2x+1) dx


The Attempt at a Solution



let u= e(x^2 +x)
du=e(x^2 +x)(2x+1)dx

integral e(x^2 +x)(2x+1) dx = integral 1/u du

am I on the right track? i didnt get the same answer as the prof...
 
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  • #2


If you mean

[tex]
\int e^{x^2+x} (2x+1) \, dx
[/tex]

then your substitution is one of at least two that will work, but how do you obtain

[tex]
\int \frac 1 u \, du
[/tex]

as the next step?
 
  • #3


because du=e(x2 +x)(2x+1)dx ?

integral e(x2 +x)(2x+1) dx = integral du/e(x2 +x)
=integral 1/u du
?
 
  • #4


If

[tex]
u = e^{x^2+x}
[/tex]

then

[tex]
du = e^{x^2+x}(2x+1) dx
[/tex]which is exactly the form of the original integral.
why is there need for a fraction?
 

1. What is the purpose of integrating e(x^2 +x)(2x+1) dx?

The purpose of integrating this expression is to find the area under the curve represented by the function e(x^2 +x)(2x+1). This is useful in many applications, such as calculating the work done by a varying force or determining the probability distribution of a continuous random variable.

2. What is the general process for integrating a function?

The general process for integrating a function involves finding the antiderivative, or the function that when differentiated, gives the original function. This can be done using integration techniques such as substitution, integration by parts, or partial fractions.

3. How do I approach integrating a function like e(x^2 +x)(2x+1) dx?

In this case, the best approach is to use the technique of integration by parts. This involves choosing one part of the function to be the "u" term and the other part to be the "dv" term. Then, using the formula ∫u dv = uv - ∫v du, you can find the antiderivative.

4. What are the limits of integration for e(x^2 +x)(2x+1) dx?

The limits of integration for this expression will depend on the specific problem or application. In general, the limits will be given in the form of x=a and x=b, where a and b are constants. These values represent the starting and ending points on the x-axis for the area under the curve.

5. Are there any special cases or tricks for integrating e(x^2 +x)(2x+1) dx?

Yes, there are a few special cases that can make integrating this expression easier. For example, if the limits of integration are symmetric about 0 (i.e. from -a to a), then the integral is equal to 0. Additionally, if the function can be factored into simpler terms, you may be able to use partial fractions to simplify the integration.

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