Evaluate the integral from standard from

In summary, the given integral is being evaluated using the standard form for an even function, with the given constants and limits of integration. The final answer is 2*(\frac{-2KT\pi}{4\kappa})1/2.
  • #1
osker246
35
0

Homework Statement


evaluate [itex]\int[/itex]e^([itex]\frac{\kappa*x^2}{2KT}[/itex])dx with limits of integration from -infinity to +infinity using the standard form [itex]\int[/itex]e^(-C*x2)dx = ([itex]\frac{\pi}{4C}[/itex])1/2 with limits of integration from 0 to +infinity. Note κ, k, and T are constants. In the standard form c indicates a constant. Note the function being integrated is an even function: f(x)=f(-x).





The Attempt at a Solution



Well looking at the equation I see C=[itex]\frac{-\kappa}{2KT}[/itex]. I then plug C into ([itex]\frac{\pi}{4C}[/itex])1/2giving ([itex]\frac{-2KT\pi}{4\kappa}[/itex])1/2.

My next step would be to evaluate:
2*[([itex]\frac{-2KT\pi}{4\kappa}[/itex])1/2][itex]^{+infinity}_{0}[/itex]

But I no longer have my variable x to do so, am I missing something? Is my answer simply ([itex]\frac{-2KT\pi}{4\kappa}[/itex])1/2?
 
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  • #2
Hint: exp(-Cx2) is an even function, symmetric about the y-axis.
 
  • #3
Thanks for the reply. I did see that it was an even function, that's why I changed my limits of intergration to 0 to +infinity and multiply the answer by two. But the integral of the standard function doesn't keep X as a variable. So how do I evaluate this integral with my limits of intergration? I feel like I am over looking something here.
 
  • #4
You needn't evaluate any limit. The integral you're given already takes into account the limits at infinity and 0.
 
  • #5
Ah, my mistake osker246. And yes, dextercioby is right.
 
  • #6
So does this mean my answer is 2*([itex]\frac{-2KT\pi}{4\kappa}[/itex])1/2?
 

Related to Evaluate the integral from standard from

1. What is the meaning of "Evaluate the integral from standard form"?

The phrase "evaluate the integral from standard form" refers to the process of finding the value of a definite integral, which represents the area under a curve on a graph, using the standard form of the integral. This involves substituting the limits of integration into the expression and solving the resulting equation.

2. How do you determine the limits of integration for an integral in standard form?

The limits of integration for an integral in standard form can be determined by looking at the bounds of the region being integrated over on the graph. The lower limit is typically the x-coordinate of the leftmost point on the graph, and the upper limit is the x-coordinate of the rightmost point.

3. What is the difference between evaluating an integral from standard form and evaluating it from an alternate form?

When evaluating an integral from standard form, the limits of integration are already given and the integral can be solved directly. In contrast, evaluating an integral from an alternate form may require using integration techniques such as substitution or integration by parts to simplify the expression before finding the limits and solving.

4. Can all integrals be evaluated from standard form?

No, not all integrals can be evaluated from standard form. Some integrals may require more complex integration techniques, such as trigonometric substitution or partial fractions, to solve.

5. Are there any tips for evaluating integrals from standard form more efficiently?

One tip for evaluating integrals from standard form more efficiently is to first check if the integrand (the expression being integrated) can be simplified or rewritten in a more manageable form. This can help to avoid more complicated integration techniques and make the evaluation process faster and easier.

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