Despite its rather grandeur-sounding name, this general property of any integrals is essentially just a restatement of a simple geometric fact: that the area under a graph is the same no matter whether you evaluate it from a to b or b to a.
The most useful formulation of this property is as follows:
We will sketch the geometric proof here for intuition. The proof essentially flips the graph and translate the graph back, giving us same limits and same area under the graph.
Let this curve be the graph of f(x):
Then, the integral from a to b of f(x) can be represented as the area under the graph from a to b, which is coloured in green (a is set to be 0.5 and b 1.0 in this demonstration, but they are just arbitrary choices):
If we reflect the graph about the y-axis, the function becomes f(-x). The original area is equivalent to the area under the reflected graph from -b to -a, as illustrated in the graph by the red area:
Now, if we translate the reflected graph in the positive x-direction by (a+b), the graph at x=-b will end up at x=a, and the graph at x=-a will end up at x=b. Now if we take the integral from a to b, we get exactly the original area, only reflected, as demonstrated by the red area on the right which equals the green area of the same shape but reflected:
Hence we derive our equation:
Now, how is this property useful? Let's see some of the examples of application:
If we want to evaluate this tan integral, there is no obvious method: it's not a product; the non-integer exponent prevents us from using the trigonometric identities. So let's try our new tool:
Firstly, label the integral I, and use the King property to get:
Because of the symmetry of the tan function:
The integral simplifies nicely back to its original form plus a constant, from which we can calculate its value.
So when shall we try and use this property? Is there any indicators? Seeing another example of the applications might help us to clarify this a little.
Here we have another trig integral:
The limits are again pi/2. This time, it might ring a bell of some trig equalities involving sin and cos which you might have come across:
Apply the King property and then the equalities, we arrive at a form of the integral which looks suspiciously like the original one:
Here we can use a clever trick: adding the two forms of the same integral together; now the numerator and the denominator cancel, and we get a constant independent of n:
That was a rather satisfying integral. Though what can we learn from it? Here's my summary:
We could try using the King property if:
1) If we see trig functions inside an integral
2) If the limits include pi
When evaluating the resultant integral, one should try to spot the original or add the original form.
But is the King Property's use limited to trigonometric functions? Here is another example, which apparently does not involve any trig or pi:
From the denominator one sees a hint to try tan substitution, which leads us to a simplified integral:
Now we have another integral with pi in the limits and trig function inside, so perhaps we could make use of the King Property again:
Turns out it worked!
To conclude, the King property might be useful even when there is apparently no trig function involved; when trig substitution is involved, we could still try with this property. (P.S. remember - labelling the integral as I can often be useful.)
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