Please help with numerical integration

In summary: As per Mathematica's NIntegrate.I don't know if you want to stay with the original integral or not. In any case a substitution will make this much easier.u=1/xoldown=x sin(x)/(1+x^3)dxthen since x(sin(.1)/.1)<sin(x)<x for 0<x<.1\frac{\sin(.1)}{.3}\log(1.001)=10\sin(1/10)\int_0^\frac{1}{10}\frac{u^2}{1+u^3}du<\int_0^\frac{1}{10}\frac{u
  • #1
undefined83
8
0
Im supposed to solve
integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way.

P.S. sorry for bad spelling and lack of proper formula notions.
 
Physics news on Phys.org
  • #2
I process from integral i get
1/3[ln((x+1)/sqrt(x^2-x+1))+sqrt(3)*arctg(2*sqrt(3)*x/3-sqrt(3)/3)] from M to +infinity <=1/4*10^-2

than i get

-ln((M+1)/(sqrt(M^2-M+1))) +sqrt(3)*(pi/2-arctg(2*sqrt(3)*M/3-sqrt(3)/3))<=7.5*10^-3

and i can't find any exact solution to solve that
 
  • #3
undefined83 said:
Im supposed to solve
integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way.

P.S. sorry for bad spelling and lack of proper formula notions.
break the interval up
10=x0<x1<x2<...<xn-1<xn=infinity
for
int(10,x1)
int(x1,x2)
.
.
.
int(xn-2,xn-1)
use simpsons rule
for
int(xn-1,infinity)
chose xn-1 large
estimate the integral on each subinterval accurate enough so that total error is within limit
 
  • #4
Thanks, but i am totaly lost when i need to find int(xn-1,infinity). I try to make it lower than error estimate but i get either logaritmic inverse-trigonometric inequality as above or another insolvable integral.
 
  • #5
undefined83 said:
Thanks, but i am totaly lost when i need to find int(xn-1,infinity). I try to make it lower than error estimate but i get either logaritmic inverse-trigonometric inequality as above or another insolvable integral.
Don't over think it
[tex]|{\int_x^\infty \frac{\sin(\frac{1}{x})}{1+x^3}}|<\int_x^\infty \frac{1}{x^3}dx=\frac{1}{2x^2}[/tex]
 
Last edited:
  • #6
Thank you very, very much. You are a life savior.
 
  • #7
Does someone who has Maple or Mathlab can give me aproximate solution of the integral, posiblly of int(10,45), and int(10,142). Error comarision goes to absurd. I get 3.3*10^-4 on pocket calculator for int(10,35). Simpson with above given boundries gives more than 10 times larger values.
 
  • #8
undefined83 said:
Does someone who has Maple or Mathlab can give me aproximate solution of the integral, posiblly of int(10,45), and int(10,142). Error comarision goes to absurd. I get 3.3*10^-4 on pocket calculator for int(10,35). Simpson with above given boundries gives more than 10 times larger values.

[tex]\int_{10}^{45} \frac{Sin(1/x)}{1+x^3}dx\approx 0.000329176[/tex]

[tex]\int_{10}^{142} \frac{Sin(1/x)}{1+x^3}dx\approx 0.000332717[/tex]

As per Mathematica's NIntegrate.
 
Last edited:
  • #9
I don't know if you want to stay with the original integral or not. In any case a substitution will make this much easier.
u=1/x
[tex]\int_{10}^\infty \frac{\sin(\frac{1}{x})}{1+x^2}dx=\int_0^\frac{1}{10}\frac{u\sin(u)}{1+u^3}du[/tex]
then since x(sin(.1)/.1)<sin(x)<x for 0<x<.1
[tex]\frac{\sin(.1)}{.3}\log(1.001)=10\sin(1/10)\int_0^\frac{1}{10}\frac{u^2}{1+u^3}du<\int_0^\frac{1}{10}\frac{u\sin(u)}{1+u^3}du<\int_0^\frac{1}{10}\frac{u^2}{1+u^3}du=\frac{1}{3}\log(1.001)[/tex]
The average of these yeild a good approximation.
simpsons rule will meet the error tolerance
I~(.1-0)/6(f(0)+4f(.05)+f(.1))
where f(x)=x sin(x)/(1+x^3)
 
Last edited:
  • #10
Thank. This afternoon i finaly finished the monster with int(10,150) by Simpson. Got aprox 0.000324. I can't really believe its finished.
 

1. How do I perform numerical integration?

Numerical integration is a method used to approximate the definite integral of a function. It involves dividing the area under a curve into smaller sections and using mathematical formulas to calculate the total area. This can be done using techniques such as the trapezoidal rule or Simpson's rule.

2. Why is numerical integration important in science?

Numerical integration is important in science because it allows us to approximate the values of integrals that cannot be solved analytically. This is particularly useful in fields such as physics, engineering, and economics, where complex equations and real-world data often require numerical methods for integration.

3. What is the difference between numerical integration and analytical integration?

Analytical integration involves finding the exact solution to an integral using mathematical techniques, while numerical integration involves approximating the solution using numerical methods. Analytical integration is often preferred when possible, but numerical integration is necessary for more complex or non-analytical functions.

4. What are some common challenges in numerical integration?

One common challenge in numerical integration is choosing an appropriate method for the given function and data. Some methods may be more accurate or efficient for certain types of functions. Additionally, rounding errors and the choice of step size can also affect the accuracy of the result.

5. How can I improve the accuracy of my numerical integration?

There are several ways to improve the accuracy of numerical integration, including using a more precise method, decreasing the step size, and increasing the number of data points. It is also important to carefully consider the limits of integration and any potential sources of error in the data. Additionally, using computer software or programming languages for numerical integration can often lead to more accurate results.

Similar threads

  • Introductory Physics Homework Help
Replies
28
Views
221
Replies
5
Views
615
  • Programming and Computer Science
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
872
  • MATLAB, Maple, Mathematica, LaTeX
Replies
27
Views
3K
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
Replies
8
Views
1K
  • Programming and Computer Science
Replies
15
Views
2K
  • Computing and Technology
Replies
5
Views
1K
Replies
3
Views
1K
Back
Top