Fig. 1

In my previous blog “A Case of Pre-FTC Definite Integration“, we obtained result

without the Fundamental Theorem of Calculus.

Let’s now consider the case of where . Namely, , the area under the curve from 1 to .

Closed form result (1) is not applicable since .

Attempt of finding the limit of a sum quickly bites the dust too due to the fact that .

However, let

We see immediately that , i.e.,

Other properties of function can be extracted from (2), as shown below:

By definition,

,

and

.

Therefore,

Let , we have

,

i.e.,

By (3-2),

,

i.e.,

Let ,

When ,

.

Assume when where , we have

Hence,

Moreover, where ,

As result,

With (3-4), (3-5) and (3-6), we conclude that

We will leave this post with the following observation:

Fig. 2

This is not difficult to see. , if , then from Fig. 2, we have

area in blue .

The fact that

area in blue

means

,

or,

,

Since ,

from which (4) is obtained.

When , area in blue is

.

Fig. 3

From Fig. 3 we see that

.

Hence

,

from which (4) is obtained again.

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