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 . A very common method of integrating this function is to make the substitution . Moreover, . Hence, we get .
Today we have to understand why that works.
When we say , what we’re saying is for some variable , every number in the interval is the image of some value of . In others words, there is a bijective mapping . Hence if we replace some value of with the sine of the corresponding , we should not find a difference.
However, there is the matter of . The function is nothing but the limit of making the intervals on the smaller and smaller, and finding the summation (of course this limit is valid only if the same limit is reached regardless of the we take in the interval). Now as , we have for some , by the Mean Value Theorem. Moreover, the limits of integration will also change, but this is obvious.
We ultimately get .
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 . We can write this very summation in terms of another function (substitution for . That is all that we’re doing here.
I hope this article helps those who ask “why”, and suffer for it.