3 Challenging Integration problems

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In summary: For the last one, let's try this:\int_{-\infty}^{\infty} e^{-ax}dx = \surd \frac{\pi} a a>0You can use the divergence theorem to determine that\int_{-\infty}^{\infty} e^{-ax}dx = -\frac{\pi}{2}a for all real values of a.
  • #1
jimmychim
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Homework Statement


1. Show that [tex]\int_0^\infty x^{n}e^{-ax}dx = \frac{n!} {a^{n+1}}[/tex]
for n = 0, 1, 2, 3...

2. Show that [tex]\int_{-\infty}^\infty x^{2n}e^{-ax^{2}}dx =\frac{{\surd \pi} (2n-1)!} {2^{n}a^{(2n+1)/2}}[/tex]
for n = 0, 1, 2, 3...

Assumption: [tex]\int_{-\infty}^\infty e^{-ax^{2}}dx =\surd \frac{\pi} a [/tex]
a>0

3. Evaluate [tex]\int {\frac{1} {A^{x^2}+Bx+C}} dx [/tex]
For all possible real values of A, B, C.

For #1 and #2, you may use mathematical induction, if you like.

Notation: 7! = 7 * 5 * 3 * 1

Homework Equations


The Attempt at a Solution

 
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  • #2
1) Use gamma function or integration by parts
2) Use gamma function (or induction)
3) Seperate the problem in certain cases.
 
  • #3
Since you post them here, I assume that you want to solve these yourself and haven't posted them as challenge for us :)

For the first one, try partial integration.
For the second one, try differentiating [tex]\int e^{-a x^2}[/tex] (this is a familiar integral in physics).
For the third one, I have no idea yet (but solve 1 and 2 first :biggrin:)
 
  • #4
i actually would appreciate if you can finish the challenge
gosh, I've spend so much time figuring these questions out.
but i just can't get thru 'em
 
  • #5
I would have appreciated it if someone would have finished my questions...especially without trying it myself...
 
  • #6
jimmychim said:
i actually would appreciate if you can finish the challenge
gosh, I've spend so much time figuring these questions out.
but i just can't get thru 'em

Any one can figure out challenge questions to give someone but not be able to get through them. So don't "gosh" at us and take some advice.
 
  • #7
It's not hard, we almost gave you the answer, you just have to do the algebra now.
For example, let's look at the first one:

[tex]
\int_0^\infty x^{n}e^{-ax}dx = \frac{n!} {a^{n+1}}
[/tex]

You can work from here and check what happens if you integrate by parts all the time (just write it out 2 or 3 times and you will see the pattern).

But let's just set up a nice formal proof by induction.
First check that it works for n = 1 (that's kinda trivial, just don't forget the boundary term!)
Then suppose that
[tex]
\int_0^\infty x^{n}e^{-ax}dx = \frac{n!} {a^{n+1}}
[/tex]
and try to integrate

[tex]
\int_0^\infty x^{n + 1}e^{-ax}dx
[/tex]
by parts (hint: write [tex]x^{n+1}e^{-ax} = x \cdot (x^{n} e^{-ax})[/tex])
 
  • #8
For the first one I much prefer differentiation to integration by parts: start with

[tex]\int_{0}^{\infty} dx~e^{-ax},[/tex]

evaluate that, and start differentiating both sides with respect to a to see the pattern arise. If you actually need a formal proof, just use induction as suggested above, but I would differentiate the expression again instead of doing it by integration by parts. Doing it this way avoids the annoying boundary terms from integrating. =)

Of course, if you're allowed to just assume the answer straight away and then use induction, just use the differentiating method straight away in the induction proof without doing the differentiation to 'discover' the pattern first.
 

1. What is the concept of integration in science?

Integration is a mathematical process used in science to find the area under a curve or the accumulation of a quantity over a given interval. It involves finding the antiderivative of a function and evaluating it at two points to determine the change or accumulation.

2. How can integration be used to solve complex scientific problems?

Integration can be used to solve complex scientific problems by providing a way to calculate the change or accumulation of a quantity over a given interval. This is useful in many fields of science, such as physics and biology, where finding the area under a curve or the accumulation of a substance is necessary for understanding and predicting natural phenomena.

3. What are some common challenges faced when solving integration problems?

Some common challenges in solving integration problems include correctly identifying the function to be integrated, determining the appropriate limits of integration, and choosing the appropriate integration technique. Integration can also be challenging when dealing with complex or multi-variable functions.

4. How can one improve their integration skills?

Improving integration skills requires practice and familiarity with different integration techniques, such as u-substitution and integration by parts. It can also be helpful to review basic algebra and trigonometry concepts, as they are often used in integration problems.

5. Are there any alternative methods to integration for solving scientific problems?

Yes, there are alternative methods to integration, such as numerical integration and computer simulations. These methods can be useful when dealing with complex or multidimensional functions that are difficult to integrate analytically.

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