Approximating Integral w/ Singularity at x=0 Using Trapezoidal Rule

[sara_87]

Homework Statement

I want to approximate the following integral:

$$\int^{1}_{0}\frac{1}{x}dx$$

using the trapezoidal quadrature rule.
However there is a singularity at x=0.

Homework Equations

trapezoidal method:

$$\int^{a}_{b}f(x) dx \approx \frac{h}{2}(f(x_0))+h\sum_{i=1}^{n}(f(x_i))+\frac{h}{2}(f(x_n))$$

where x_i = i*h and h=1/n and i = 0,1,2,...,n

The Attempt at a Solution

$$\int^{a}_{b}f(x)dx \approx \frac{h}{2}(1/x_0)+h\sum_{i=1}^{n}(1/x_i)+\frac{h}{2}(1/x_n)$$

i can implement the second and last term but not the first term due to the singularity (at x=0)
How can i deal with the singularity?
any help or ideas will be very much appreciated.
Thank you.

 

[Dick]

It's worse than just the function having a singularity. The singularity is bad enough that the integral doesn't exist. In some sense, the answer to the integral is +infinity. There is an infinite amount of area under the curve. How can you (or why would you want to) approximate that?

 

[sara_87]

thank you.
But, If i was to use the trapezoidal method in this case, or in any other case where i have a singularity in the function at the end point, how would i get rid of the singularity?

 

[Dick]

thank you.
But, If i was to use the trapezoidal method in this case, or in any other case where i have a singularity in the function at the end point, how would i get rid of the singularity?

There's no way to 'get rid of it'. I think numerical methods for integrating functions with singularities are complicated. You could always make a small interval around the singularity and throw it away and apply the trapezoid method to the rest. It would be an estimate anyway.

 

[sara_87]

thank you :)
i will consider this.