Further Calculus

Introduction

In this calculus Learn page we will look into building on the differentiation and integration techniques learnt in previous lessons. We will investigate using trigonometric identities to simplify integrals to make them feasible, and also use general forms for the derivatives of inverse trig functions to further expand our "integrable" functions library.

Derivation 

 

Using Trig Identities to simplify Integrals

Take for example the following integral:
The techniques we know of (such as by-parts and substitution) wouldn't work in this case. Take for example integration by substitution. Let u=sinx, as you would, but you would get a rogue unwanted cos x as a coefficient! :
This can happen with a whole bunch of integrals of trigonometric functions. So to avoid this problem it is quite simple and elegant to use a trigonometric identity to re-write the integral. In the case of the example above:
Now that the integral can be re-written it is much more simple to get to the result:
A similar identity exists to simplify integrals involving squares of the cosine function:

How are the Identities Derived?

Using Inverse Trig Functions

There are many integrals that can seem very difficult, but can be easily spotted to be in the form of the derivative of an inverse trigonometric function. But first we need to take the derivative of the inverse trig functions.
It may not seem so obvious at first but:
We can now take the derivative of both sides with respect to x (implicit differentiation) to help us find the derivative of arcsin:
The same technique can be used to work out similar results for arccosx and arctanx:
Therefore we can make general results for integrals in these forms:

Trig Substitutions

You have for sure come across integration for substitution by the time you are studying these calculus techniques- if not make sure you visit the PlanckTime Learn page on integration by substitution. We can further expand the "integrable functions" library by ...
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