How can you know what substitution you have to do for $\int \sec x\, dx = \ln |\sec x + \tan x| + C$

Question

I've seen many videos calculating this integral: $\int \sec x\, dx = \ln |\sec x + \tan x| + C$

But I've never understood why you multiply and divide by $\sec x + \tan x$
And then substitute $\sec x + \tan x$ by $u$

How can I think about this? Is there some rule or something because I wouldn't have thought about doing that. Is evaluating integrals just memorizing something someone has already done?

Answer 1

Great Question! We multiply and divide the integral by $\sec(x)+\tan(x)$ in order to make the numerator the derivative of the denominator. But of course it would not have come to mind on first thought. There is no need to memorize, as we could have taken any one of the several methods to solve this integral. We could also multiply and divide by $\sec(x)$, and use the identity $\sec x = \sqrt{1+\tan^2x}$ in the denominator, and substitute $\tan(x)$ by a variable, say $z$. The numerator becomes $dz$ and the denominator $\sqrt{1+z^2}$ then we solve using the standard formula.

Answer 2

Here is another "thinking outside the box" method you won't find in the books: Add and subtract $tanx$. Your first integral is $\int\frac{1+sinx}{cosx}dx$. Multiply top and bottom with $cosx$, apply the Pythagorean theorem in the denominator, factor, cancel and you get a very easy integral to integrate (You do the steps!). Second integral is $-\int{tanx}dx$ and that is a natural log. Combine your results, do a little trig algebra and done! Same strategy can be done for $\int{cscx}dx$ where you add and subtract $\int{cotx}dx$. An exercise for you. And here is another approach. Multiply top and bottom with $cosx$. Apply the Pythagorean Theorem on the denominator and do a u-sub followed by Partial Fraction Decomposition. Very easy.