Wonders of Integral Calculus

“If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing” ~ Henri Poincare

It’s been a long time since I’ve published something. However this article is really special to me and many others who discovered Calculus through sneaky explorations and fell in love with Math.

If you’ve never done any integral calculus problems before, I suggest you look through Thomas’ Calculus and Apostol’s Calculus and get the prerequisites for this article.

Now let me cut to the chase and show you the first problem.

The problem at first looks really difficult. But once you look at the idea you may feel the same way I did: this problem is one of the easiest integrals!

Now the idea involves an understanding of Riemann’s integration. You can watch the video below to get an understanding-

Khan academy: Video for Introduction to Integral Calculus

Now once you think of the definite integral as a Riemann sum, things become much easier. But before I start doing the problem directly I’ll show you a result related to even/odd functions:

Let f(x) be any function. We can write f(x) in an alternate way-

And use this to compute our integral above. Letting f(x) be the integrand-

And Voila! We have found what we set out to. This is why I said that it would become one of the easiest integrals ever if you just did the necessary manipulations. The idea was to represent a function as the sum of an even and odd function- that you can do such a manipulation to any function is a testament to the beauty of Math. And you could do the above trick for any f(x) instead of the e^(1/x) in the denominator as long as f(x)f(-x) = 1. Also if we had integrate the same integrand from -A to A for all A in the domain of f we would always get sin(A), which is something fantastic in itself.

Now let’s look at another seemingly ‘difficult’ integration problem that becomes easy with some manipulations.

Wow! Such a complicated function to nothing. Literally nothing. You could look at the graph and get the intuition that the areas bounded by the x-axis and graph below and above the point (1,0) are equal. Still wonderful!

Graph for y = lnx/(1+x²)

We can generalize this by applying the same trick to the more general case-

Generalization of the integral above

I suggest you try this definite integral problem with a cool trick of its own-

Hint: Partial Fractions

(Note the singularities of the integrand- asymptotes in graph of 1/(x³-1)- try taking limits of the indefinite integral before integrating it over the interval)

Now that we have done some really fascinating but easy definite integrals, let me show you something you can do with just the knowledge of integration techniques.

First consider integration by parts. We use this technique when the integrand appears to be the product of two functions whose derivatives we know. But we also use the technique in the case when the integrand is difficult to integrate.

Consider this: Wikipedia- ln x integration

But what if we do a similar thing for any function f(x)? Integration by parts states-

I leave proving the conjecture up to you. I proved it using First principle of Finite induction since it is conjectured to be true for all positive integers n.

But what happens if we continue integrating by parts again and again? Let us limit the above expression on right-hand side with n approaching infinity-

Woah!

And we have shown the integral as a power series! Just note the integration constant C that I added at last. It is necessary for the indefinite integral for a general answer.

This brings me to the conclusion of this article. Please go check out my other articles on the main website and go read ‘Inside Interesting Integrals’ by Paul Nahin.

Clap, comment, share and follow my blog Cosmus Mathematicus!