The Beauty of Integration
By Avi Banerjee
(Physics, Biology, Chemistry, Math, History Tutor at The Edge Learning Center)
My old maths teacher from school had introduced me to integration by showing the beauty of mathematics. Integration, she informed me, is an art, requiring creativity and ingenuity. Unlike differentiation, where there are pre-set rules be it the product rule or chain rule, integration requires one to examine the integrand and use one’s mathematical practise to find the solution. There are no predefined instruction on what substitution to use during integration via substitution, nor are there any given instructions of what to set as for by parts. Yet, this is not to mean that the art of solving integrals is random or cavalier, on the contrary, it requires mathematical intuition. Integration by substitution will only work, when the right function is substituted, and an appropriate substitute is given. And in many instances, there will not be any way to use substitution to solve the integral. Similarly for by-parts, there are limitations on when it can be used, and when it is used, there are often only one correct method requiring an appropriate choice in . Let me give you an example:
The integral above is an unknowable integral, where conventional by parts or substitution will not yield a direct result. For starters, there are no obvious substitutes one can use, because both are not derivatives of one another, thus do not simplify or get reduced as they undergo differentiation. This would mean, of the methods taught in the IB HL Mathematics course the only technique left is by parts. Let’s see what happens when we use by part, and take
Now, one can clearly see the problem:is similar to the original question at hand, as it involves a trigonometric function and an exponential function.
Before we move ahead, I must give a disclaimer. Whilst the original question is technically an unknowable integral, we do know how to solve it. Yes, I know that might sound paradoxical, but through the beauty of mathematical logic, we can pass this hurdle. To begin with, had we set , this problem would have been truly unsolvable. It was important to set is the same as This property, we will see will be vital for the solution.
To continue, one must employ a second by parts, for the integral by setting
Now, by inserting equation (3) into equation (2), an interesting occurrence occurs:
One can see that the term, which was the original question, appears on both sides of the equation. By setting , and transferring , to one side of the equation, one is able to find the solution without actually knowing the integral:
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