How do you Find the Summation of the area under the function f(x) = 1/x when n is not equal to infinity

Question

kesumonu · Answer

could you specify what is n in this case?

gerywahyu · Answer

I have a hard time understanding this question. But I think what you mean is the are under the function. So the area of the function $$f(x) = 1/x$$ is equal to the integral of the function. $$\int\limits_{b}^{a} f(x) dx$$
Put the function there
$$\int\limits_{b}^{a} \frac{ 1 }{ x } dx$$
And integrate it
$$\ln x$$

emperorpalpatine · Answer

Ah, good question. As you may have seen with your studies before of calculus, this method of finding anti-derivatives:
$$\int\limits_{}^{}x ^{n}=x ^{n+1}\div n+1$$

However, note what happens when we apply the same function where in this case n is a negative integer lets say -1. ( Note x to the -1 is the same as 1 over x, so don't get confused.)

$$\int\limits_{}^{}x ^{-1}dx=x ^{-1+1}\div -1+1$$

You see that we are then dividing by 0! This is not allowed in mathematics!
But we know that 1 over x is indeed a function and that it has horizontal and vertical asymptotes. 
So the question arises on how to find an anti-derivative of this function in which you then plug in your limtis of integration and subtract to find an area or a numerical value. But please make sure to not have ur limits of integration for this particular function as 0, negative infinity, and positive infinity, as you will end up with an improper integral which you will learn further on in your studies about.

But back to the case of finding this anti-derivative. We define the antiderivative of this function to be as follows:

$$\int\limits_{}^{}x ^{-n}dx=\ln x$$ 

You will see a proof of this in your calculus textbook, which I strongly encourage you to do, to understand the purity of this conjecture.

AND REMEMBER: all of these integrals i have shown above are INDEFINITE integrals. If you had any difficulty understanding this I suggest you review your antiderivatives. 

also remember that the area is simply an application of these antiderivatives you are calculating with these integrals.

 I am a Grade 9 student so if I made any minor spelling miztakes, then just live with it.