There are multiple ways of integrating the secant function, and some of them are quite advanced. For instance, it is possible to get the answer in complex form, or inverse hyperbolic form. For now, we'll just look at 2 of the more "elementary" ways of solving the integral of sec(x), i.e. ...
...by:
- Magic manipulation or pattern recognition (substitution)
- Partial Fraction Decomposition
1. Magic manipulation or pattern recognition
This is not my preferred method because it sort-of requires you to know the result that you're trying to find, which is a bit "ass backwards".
But it's great for those who are good at recognising patterns in mathematics and have that brilliant intuition that we all lust after...
Anyway, the idea with this is to manipulate the integral into the form...
To do this, we can recognise that derivatives of sec(x) and tan(x) are:
So, let's make...
So, what if we manipulate the integrand sec(x) as follows...
Effectively, we are multiplying sec(x) by 1, right?
Now, multiplying sec(x) into the numerator, we can carry through with the operation...
So there you have it - the integral of sec(x). We'll look at method 2 next and see how the results compare.
If you need a clearer explanation, play my video below...
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