Need help solving what seems like a strange integral

In summary, the conversation discusses finding the Fourier series for the function f(x)=|x|^3 on the interval -L<x<L. It is determined that the function is even, so it will be a cosine series. The process involves using the piecewise expansion of |x| and performing integration by parts. The final result is determined to be A_n=\frac{2}{L}\int_{0}^{L}x^{3}cos{\frac{n\pi x}{L}}dx and A_{0}=\frac{1}{L}\int_{0}^{L}x^{3}dx.
  • #1
saybrook1
101
4

Homework Statement


I need to find the Fourier series for [itex]f(x)=|x|^3[/itex].

Homework Equations


[itex]f(x)=|x|^3[/itex]
[itex]A_n=\frac{2}{L}\int_{0}^{L}|x|^3cos{\frac{n\pi x}{L}}dx[/itex]
[itex]\int_{0}^{L}x|x|sin{\frac{n\pi x}{L}}dx[/itex]

The Attempt at a Solution


Since it is an even function I know that it will be a cosine series and so I set out to find the A coefficient like so:
[itex]A_n=\frac{2}{L}\int_{0}^{L}|x|^3cos{\frac{n\pi x}{L}}dx[/itex]

Through integration by parts I end up with a sine term that goes to zero and some coefficients out in front of an integral that looks like:
[itex]\int_{0}^{L}x|x|sin{\frac{n\pi x}{L}}dx[/itex]

I would really appreciate any help with this particular integral or if someone could point me in the direction of a solution where a Fourier series is calculated for a function with an absolute value to the power>2. Thanks in advance.
 
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  • #2
When dealing with |x| it often helps to use the piece wise expansion of it:
=x, x>0
= -x, x<0.
If you were on the interval from [-L,L] you would have 2 integrals one from [-L,0] and one from [0,L]. However, since you are only interested in the function from [0,L] then $|x|^3$= $x^3$. To solve the integral you have integration by parts like normal.
 
  • #3
Did you mean [itex]\int_0^L |x|^3 cos(\frac{n\pi x)}{L}dx[/itex] of [itex]\int_{L/2}^{Ll/2} |x|^3 cos(\frac{n\pi x)}{L}dx[/itex]? In the first, which is what you wrote, x is always non-negative so |x|= x. For the second, do the integrals from -L/2 to 0 and from 0 to L/2 separately.
 
  • #4
HallsofIvy said:
Did you mean [itex]\int_0^L |x|^3 cos(\frac{n\pi x)}{L}dx[/itex] of [itex]\int_{L/2}^{Ll/2} |x|^3 cos(\frac{n\pi x)}{L}dx[/itex]? In the first, which is what you wrote, x is always non-negative so |x|= x. For the second, do the integrals from -L/2 to 0 and from 0 to L/2 separately.
Yeah, that was my bad; the question wants me to find the Fourier series for f(x)=|x|^3 on -L<x<L. So I think I can choose [itex]A_{n}=\frac{2}{L}\int_{0}^{L}x^{3}cos{\frac{n\pi x}{L}}dx[/itex] and then [itex]A_{0}=\frac{1}{L}\int_{0}^{L}x^{3}dx[/itex]?
 
  • #5
Jufro said:
When dealing with |x| it often helps to use the piece wise expansion of it:
=x, x>0
= -x, x<0.
If you were on the interval from [-L,L] you would have 2 integrals one from [-L,0] and one from [0,L]. However, since you are only interested in the function from [0,L] then $|x|^3$= $x^3$. To solve the integral you have integration by parts like normal.
Thanks for your reply. I was unclear in the question statement; It asks me to find the Fourier series for the function f(x)=|x|^3 on -L<x<L. I guess I can choose [itex]A_{n}=\frac{2}{L}\int_{0}^{L}x^{3}cos{\frac{n\pi x}{L}}dx[/itex] and then [itex]A_{0}=\frac{1}{L}\int_{0}^{L}x^{3}dx[/itex]?
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It can also be thought of as the accumulation or total of a function over a certain interval.

2. How do I solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric substitution. You can also use online integral calculators to get the answer.

3. Why does this integral seem strange?

Some integrals may seem strange because they involve complex mathematical concepts or require advanced techniques to solve. They may also have unusual limits or integrands.

4. Can you give an example of a strange integral?

One example of a strange integral is the Fresnel integral, which involves complex numbers and has no closed-form solution. Another example is the integral of the Gaussian function, which is used in statistics and has a unique solution.

5. Where can I get help with solving integrals?

You can seek help from math tutors, online forums, or your classmates. You can also use online resources such as integral tables, calculus textbooks, or instructional videos to learn how to solve integrals.

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