Integral Help: Solve \frac {2(1+x)} {1-2x-x^2} dx

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In summary, the conversation involves a person seeking help with an integral and another person suggesting the use of substitution and completing the square to find the solution. The integral is eventually simplified to ln(1-2x-x^2) and the conversation also includes a question about a similar integral involving (x+1)^2.
  • #1
Sparky_
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Greetings

Can you help with the following integral:

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

I'm reasonably sure my setup is correct up to this integral. I tried to factor and do some canceling. - no luck'

thoughts and direction

Thanks
-Sparky-
 
Last edited:
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  • #2
Have you tried substitution?
 
  • #3
Put, x^2+2x-1=z
 
  • #4
AHHH! thanks -

[tex] u = (1-2x-x^2) [/tex]
[tex] du = -2 - 2x dx[/tex]
[tex] dx = \frac {du} {-2(1+x)} [/tex]
[tex] -\int \frac {-du} {u} [/tex]

[tex] = ln(1-2x-x^2) [/tex]

This solution is in the exponent of "e"

and leads to the integral below.
Question: can you suggest a start for:

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

I've tried various substitutions again and don't see it.

I've tried [tex] u = -x^2 - 2x [/tex]
[tex] du = -2x - 2 dx [/tex]
or
[tex] (-2(x+1) )dx [/tex]
[tex] dx = \frac {du} {-2(x+1)}[/tex]

leaves me with a (x+1) term

thanks
Sparky_
 
Last edited:
  • #5
Note that [tex]1-2x-x^2 = -(x+1)^2 + 2[/tex].

Now separate, and integrate.

This is completing the square. Also you could multiply the bottom out and long divide to get a similar result.
 

1. What is "Integral Help: Solve \frac {2(1+x)} {1-2x-x^2} dx"?

"Integral Help: Solve \frac {2(1+x)} {1-2x-x^2} dx" is a mathematical expression known as an integral, which represents the area under a curve on a graph. In this specific case, the expression is asking for the integral of the function \frac {2(1+x)} {1-2x-x^2} with respect to x.

2. Why is it important to solve integrals?

Solving integrals is important in mathematics because it allows us to calculate the area under a curve, which has many real-world applications such as finding the distance traveled by an object with varying velocity or determining the volume of irregular shapes.

3. What is the process for solving this integral?

The process for solving this integral involves using techniques such as substitution, integration by parts, or partial fractions to manipulate the expression into a form that can be easily evaluated. Then, we use the fundamental theorem of calculus to find the antiderivative of the function and evaluate it at the given limits.

4. Are there any special rules or formulas that can help solve this integral?

Yes, there are special rules and formulas for solving integrals, such as the power rule, trigonometric substitutions, and u-substitution. It is important to identify which technique will work best for a given integral.

5. Is there a specific method for checking the accuracy of the solution?

Yes, there are several methods for checking the accuracy of a solution to an integral, such as taking the derivative of the antiderivative to see if it matches the original function, using a graphing calculator to plot the function and the integral, or using numerical integration techniques to compare the value of the integral to the actual area under the curve.

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