Trigonometric substitution in integration

Why trigonometric substitution in integration: this is something that puzzled me and made me hate differentiation/integration during my IITJEE preparation. A lot of the techniques that we learned were based on algorithmic memorization rather than a feel for what really was happening. Thankfully, I have come to a college that requires 0% attendance, so that I can fulfill all my deepest desires for understanding.

Say we have $\int_{0}^{1}{\frac{1}{\sqrt{1-x^2}}dx}$ . A very common method of integrating this function is to make the substitution $x=\sin\theta$ . Moreover, $dx=\cos\theta d\theta$ . Hence, we get $\int_0^{\frac{\pi}{2}}{\frac{\cos\theta }{|\sin\theta|}d\theta}$ .

Today we have to understand why that works.

When we say $x=\sin\theta$ , what we’re saying is for some variable $\theta$ , every number in the interval $[0,1]$ is the image of some value of $\theta$ . In others words, there is a bijective mapping $\sin\theta:\theta\to[0,1]$ . Hence if we replace some value of $x$ with the sine of the corresponding $\theta$ , we should not find a difference.

However, there is the matter of $dx$ . The function $\int{f(x)dx}$ is nothing but the limit of making the intervals $\Delta x$ on the $x-axis$ smaller and smaller, and finding the summation $\sum{f(x)\Delta x}$ (of course this limit is valid only if the same limit is reached regardless of the $x$ we take in the interval). Now as $x=\sin\theta$ , we have $\Delta x=|x_i-x_j|=\cos\theta\Delta\theta$ for some $\theta\in[\arcsin{x_i},\arcsin{x_j}]$ , by the Mean Value Theorem. Moreover, the limits of integration will also change, but this is obvious.

We ultimately get $\int_0^{\frac{\pi}{2}}{\frac{\cos\theta }{|\sin\theta|}d\theta}$ .

What we have in essence is a sequence of infinite summations, approaching a limit (the integral). Say one of the terms of the sequence is $\frac{1}{2}*\frac{1}{2}+\frac{1}{4}*\frac{1}{4}+\dots$ . We can write this very summation in terms of another function (substitution $x$ for $\cos\theta$ . That is all that we’re doing here.

I hope this article helps those who ask “why”, and suffer for it.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush View more posts

2 thoughts on “Trigonometric substitution in integration”

Even I study in an institute that requires 0% attendance but never realized that I could use this to explore and increase my level of understanding

ayushkhaitan3437 says:

May 13, 2014 at 12:58 am

Peshal, which college are you from?

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2 thoughts on “Trigonometric substitution in integration”

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