Saturday, August 4, 2012

Integration Using Wolfram Alpha

Wolfram Alpha is a web based tool that allows you carry out quite sophisicated mathematical operations, producing both analytical and graphical answers. The operations of Wolfram Alpha are based on the software "Mathematica" that was developed in the late 1980's by Stephen Wolfram and his team. The web tool is much more forgiving than the off line software in terms of syntax, though Mathematica is much more powerful and can be used as a sophisicated programming language.

However, Wolfram Alpha is simple to use, performs most mathematical opeations relevant to undergraduate and High School students and is FREE !!! 

To illustrate this great utility, head to, think of a mathematical operation and type in your question.

I typed in "Integrate cos(x) from x = -pi to pi" into the input box and pushed enter. If you don't feel so game as me, you can go to the example page where there are numerous examples of mathematical operations that can be performed and the input forms preferred by the program. If you guess the form, like I often do, the program will do its best to make sense of your crude mathematical jottings,

A few micro seconds later, the following output came back:

(Images from www.

The graphs generated by Wolfram Alpha are good quality and can be saved as PDFs. I also think the graphical nature of the solution is most helpful in visualising the mathematical operation being performed. For example, the symmetry of cos(x) integral around the y axis is quite evident in the solution above. This property means that we classify cos(x) as an "even function". A parabola is another example of an even function. "y=x" and "y=sin(x)" are simple examples of functions that don't have this symmetry, which I will discuss in more detail in a later entry.

Of course, this problem is too simple, now I type in a problem that is a little more challenging:

"Integrate x^2 cos(x)"

Wolfram Alpha eats up problems like this (this one would take me several minutes using pen and paper and even a a few seconds using my well worn integral tables) and even provides the solution steps, which is an invaluable to any student (or even a rusty Professor) trying to get on top of the mysteries of integration.

and another useful graph

(Images from

Once again, note how the graphical representation of the function makes it immediately clear that the function has a particular symmetry.

Wolfam Alpha has it critics and Stephen Wolfram himself is a controversial figure but personally I am most grateful to have access to such a obviously useful tool.

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