The integral of sec(x) - Part 2

In my last post, we worked through how to integrate the secant function using a bit of mathematical "trickery". I mentioned it's not my preferred method because it requires you to know part of the result before you start.

For those of us who do not have that level of intuition, the second approach, using partial fractions allows us to use more of our deductive reasoning processes.

So let's have a go...

2. Method of Partial Fractions

Now, if we recognise that the integrand...

The creative part of this method is to multiply the right-hand side by cos(x)/cos(x), effectively multiplying by 1...

Now, for the denominator, we can apply the Pythagorean Identity...

Now, can make the substitution u = sin(x), and integrate both sides of the following expression...

...and if we take the derivative of u with respect to x and separate the differentials, we can substitute du for cos(x)dx...

Ok great! So where does "Partial Fractions" come into play? Right about now! The integrand on the right-hand side can be rewritten as follows...

...because we have the difference of 2 squares in the denominator. Then, the fraction on the right-hand side can be decomposed into partial fractions...

Ok, after all this work, we're at a stage where we can integrate...

Finally, we need to complete the result by substituting sin(x) back in for u...

Alright, before we finish, let's compare the results that we got in method 1 versus method 2...

Are these the same? They don't appear to be, but let's see. Let's manipulate the expression inside the natural log for the second result, effectively multiplying by 1...

So equation (2) then, we can rewrite as follows...

Whew... we got there in the end. Indeed, both methods produce the same result!

If you need a clearer explanation, play my video below...

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