Basic integration trouble

In summary, the probability of finding a particle of mass m between x=0 and x=L/10, if it is in n=3 state of an infinite well, can be determined by integrating the equation P = \int_a^b\left |\psi\left(x\right)\right|^2 dx with the equation \left|\psi\right|^2 = \frac{2}{L}Sin^2\left(\frac{nx\pi}{L}\right). By using the trig identity \cos{2x}=\cos^{2}x-\sin^{2}x, the integration can be simplified and the final answer should not contain an L.
  • #1
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Homework Statement


Determine the probability of finding a particle of mass m between x=0 and x=L/10, if it is in n=3 state of an infinite well.

Homework Equations


[tex]P = \int_a^b\left |\psi\left(x\right)\right|^2 dx[/tex]
[tex]\left|\psi\right|^2 = \frac{2}{L}Sin^2\left(\frac{nx\pi}{L}\right)[/tex]

The Attempt at a Solution


I'm trying to integrate...

[tex]\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx[/tex]
step (1)
[tex]\frac{1}{L}\left[ \int_0^\frac{L}{10}1-\int_0^\frac{L}{10}Cos\left(\frac{6x\pi}{L}\right)\right]dx[/tex]step (2)

[tex]\frac{1}{L}\left[\frac{L}{10} - Sin\left(\frac{3\pi}{5}\right)\right] [/tex]

When I finish solving, I end up with an L in the answer...
which I know I'm not suppose to have, did I mess up my integration somewhere?
sorry about my pooooor pooor latex
 
Last edited:
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  • #2
[tex]\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx[/tex]

Yes?
 
  • #3
yep that's right sorry =(
 
  • #4
Sorry, PF was like broke last night. Use this trig identity:

[tex]\cos{2x}=\cos^{2}x-\sin^{2}x[/tex]

[tex]\sin^{2}x=\frac 1 2(1+\cos{2x})[/tex]

But I think that's what you did, so it's good? Sorry don't have time to work it out myself.
 
  • #5
Yes thank you!
 

1. What is basic integration?

Basic integration is a mathematical process used to find the area under a curve or the exact value of a definite integral. It is a fundamental concept in calculus and is used in many areas of science and engineering.

2. Why do I have trouble with integration?

Integration can be challenging for some people because it requires a good understanding of algebra, trigonometry, and calculus concepts. It also requires practice and a strong grasp of the fundamental principles.

3. How can I improve my integration skills?

Practice is key when it comes to improving integration skills. Make sure you have a solid understanding of the basic principles and then work on solving a variety of integration problems. You can also seek help from a tutor or utilize online resources and practice problems.

4. What are some common mistakes in integration?

Some common mistakes in integration include forgetting to add the constant of integration, missing negative signs, and using incorrect substitution or integration techniques. It is important to carefully check your work and practice regularly to avoid these errors.

5. What are some real-world applications of integration?

Integration has many real-world applications, including calculating the area under a velocity-time graph to determine distance traveled, finding the volume of irregular objects, and solving problems in physics, economics, and engineering. It is a powerful tool used in many fields of science and technology.

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