Similar thing that has always dissatisfied me is how repeated roots are handled in the characteristic equation for ODEs. The method basically boils down to “here’s the answer, check it it works.” Which is a little bit like saying “this is the integral, just differentiate and you will see”. Sure that shows the answer is correct. But it doesn’t help in deriving it. The one book that I have found that did a nice job of a natural derivation was actually 4th edition Thomas Finney Calculus.

Would note on your problem above that a lot of books give a general method with a subsitution of half angle tangent. This works in general for sin cos type problems. And the sec problem yields to it. Even here would like a little intuitive rationale for why it is the key to unlock so much.

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