Integral With Floor And Fractional Functions
Can You Solve It?
The integral shows a floor function multiplied by a fractional function. Evaluate the integral.
Solution
To start off, we should note that any real number is composed of its integer and its fractional part
For example, 3.5 = 3 + 0.5
With this, we will rewrite the fractional function as
Substituting this into our integral gives us
We will now limit the integral to the interval (n, n+1), this transforms the integral into the follows
Using the reverse power rule, we arrive at
Upon simplifying, we arrive at
As the integral ranges from x = 0 to x = 20, it’s equivalent to finding
The upper limit is 19 because, in the interval (19, 20), the floor function takes any value in the interval as 19
Computing the summation, we get
Well done!
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