Expansion of definite integral by definition ∫ (1 / N ~ 1), 1 / X * DX, n tends to infinity [(1-1 / N) / N] * (1 + 1 / 2 + 1 / 3 But why is it just equal to 1 + 1 / 2 + 1 / 3 Add 1 / n

Expansion of definite integral by definition ∫ (1 / N ~ 1), 1 / X * DX, n tends to infinity [(1-1 / N) / N] * (1 + 1 / 2 + 1 / 3 But why is it just equal to 1 + 1 / 2 + 1 / 3 Add 1 / n

The title is a bit confusing, I can't understand it

The definition of definite integral is as follows: Let f (x) be bounded in the interval [a, b]. How to explain the boundedness here? Is it not continuous in the interval?
Instead of asking about his definition, he explains why there should be a boundary?

Let the function be bounded in the upper bound and insert some points in it
Divide the interval into small areas
,
The length of each cell is
At any point in each cell, the product of the function value and the cell length is made, and the sum is given
（3）
Remember, no matter how to divide a function, no matter how to choose a point in a small area, as long as at that time and s always tend to a certain limit, then we call this limit the definite integral of a function in an interval
（4）
They are called integrand function, integrand expression, integral variable, integral lower limit, integral upper limit and integral interval

Finding definite integral ∫ [0, √ 3A] 1 / (a ^ 2 + x ^ 2)

∫[0,√3a] 1/(a^2+x^2)
=∫[0,√3a] 1/a^2(1+(x/a)^2)
=1/a^2*∫[0,√3a]1/(1+(x/a)^2)
=1/a^2*arctanx/a|[0,√3a]
=1/a^2*π/3
=π/3a^2

Finding the definite integral of e ^ [(1 / 2) x] - upper line 1, lower line 0

∫ online 1, offline 0 e ^ [(1 / 2) x] DX
=2 ∫ online 1, offline 0 e ^ [(1 / 2) x] d (x / 2)
=2e^[(1/2)x]|(0,1)=2e^(1/2)-2