How to Integrate to Infinity in Python

In mathematical analysis, one of the most common tasks is to evaluate integrals which result in infinity. However, this task can become complicated very quickly when done manually.

Thankfully, the Python programming language, with its powerful libraries such as NumPy and SciPy, can make evaluating this kind of integrals significantly easier and more straightforward. In this tutorial, we’ll get to know how to evaluate or integrate to infinity in Python.

Step 1: Import Necessary Libraries

The first step is to import the necessary Python libraries that we are going to use in this tutorial. The essential libraries for this task are NumPy and SciPy. NumPy provides support for numerical operations that include mathematical, logical, shape manipulation, sorting, and other types of operations.

The SciPy library, on the other hand, is built to work with NumPy arrays and provides many user-friendly interfaces for tasks such as numerical integration and optimization. Here is how we can import these libraries:

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import numpy as np
from scipy import integrate

Step 2: Define The Function

The next step is to define the function that you want to integrate to infinity. For the sake of this tutorial, let’s consider a simple function, for instance, f(x) = e^-x. The function can be defined in Python as follows:

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def func(x):
    return np.exp(-x)

Step 3: Solving The Integral

Now, we use the quad() function from the integrated module of SciPy to evaluate the defined infinity integral. The quad() function calculates the indefinite integral of a function from a to b (possibly infinite interval). The function takes three parameters: the function that should be integrated, the lower limit, and the upper limit. Here’s how you can use it:

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result, error = integrate.quad(func, 0, np.inf)
print(result)

Step 4: Output The Result

Finally, we should print out or output the result from the integral. This can be done with the simple Python print command:

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print(f"The integration result is {result} with error of {error}")

Full Code

Here is the complete code that integrates the function to infinity and outputs the result:

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import numpy as np
from scipy import integrate
 
def func(x):
    return np.exp(-x)
 
result, error = integrate.quad(func, 0, np.inf)
print(f"The integration result is {result} with error of {error}")
 
The integration result is 1.0000000000000002 with error of 5.842606703608969e-11

Conclusion

As we can see, the combination of NumPy and SciPy libraries in Python simplifies the process of integrating to infinity. It provides a straightforward and efficient method to perform complex mathematical analyses. This tutorial provides a step-by-step guide to integrate functions to infinity in Python.