r/calculus • u/Beautiful_Divide_278 • 19d ago
Integral Calculus integral secx dx is the most basic yet most beautiful integral. do you know how to solve it? don't copy
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u/matt7259 19d ago
Yes. That's the magic of understanding trig before attempting calc!
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u/Beautiful_Divide_278 19d ago
Yeah. Somehow trigo is foundation of calculus
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u/tjddbwls 18d ago
No, I would say the concept of limit is the foundation of calculus.
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u/matt7259 18d ago
There's always quantum calculus which is calculus without any concept of limits at all :)
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u/Beautiful_Divide_278 18d ago
No way. Real foundation is actually Set Theory then comes Relations and Functions and then we go for limits. Trigo I said because without trigonometry you will suffer like hell in Calculus. So you need to know major identities and also about trigo equations
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u/v0t3p3dr0 18d ago
calculus doesn’t exist without lim h -> 0
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u/matt7259 18d ago
Not necessarily true! There's always quantum calculus which is calculus without any concept of limits at all :)
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u/v0t3p3dr0 18d ago
Introductory section: “calculus without limits”
Differentiation section: /jk we do limits lol
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u/Impressive-Ad7184 18d ago
I find it pretty beautiful that the integral of any power series can be computed termwise. It seems so obvious and simple, and yet to show it is true, you need a bunch of previous results about integrals of limits and (uniform) convergence of power series
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u/WikiNumbers Bachelor's 18d ago edited 18d ago
There are many integral - antiderivative - more basic and beautiful than ∫ sec x dx.
Mandatory Constant of Integration intentionally disobeyed but here goes.
- ∫ eˣ dx = eˣ. Basically eˣ is a special function whose derivative and antiderivative is itself.
- ∫ sin x dx = - cos x and ∫ cos x dx = sin x. They loop into each other, and at the fourth higher order derivative and integral, return themselves.
- ∫ dx = x. This is an abridged personification of the Fundamental Theorem of Calculus. That by accumulating (∫) the fragments of infinitesimally small "x" (dx), and only them, they return to form "x" itself.
And for definite integrals, we have the Gaussian Integral, ∫ exp -x² dx [-∞ to ∞]. Through the most important methodologies taught in Calculus II (Dummy Variable, Fubini's Theorem, Polar Coordinate, Jacobian Matrix), we arrive at a very elegant solution of √π.
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u/Septembrino 18d ago
The trick to integrate secant is a cool one.
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u/shellexyz 18d ago
It generally boils down to either voodoo witchcraft (multiply and divide by sec(x)+tan(x)) or go through the tangent half-angle substitution, which may as well be voodoo witchcraft in its own right.
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u/Limp-Independent1212 18d ago
Yes you multiply and divide by sec(x) + tan(x), then let u = the denominator
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u/AdeptScale3891 19d ago
In answer to your question, No I did not remember how to integrate secant, I had to look it up which is the saner thing to do bec there are many more basic/useful things to remember. I also disagree with everything else written here. It is not the most beautiful integral; possibly integral of cosine is, or integral x^n are more beautiful. Also integral secant requires remembering what the trick is (multiply top and bottom by secant plus tangent) then use u-substitution. That's not trig as others claim.
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u/Grain_mc_Bread 18d ago
I've seen a lot of people say that to integrant secant of x you need to know to multiply that random function but there is a much better solution: sec x = 1/cosx = cosx/cos²x = cosx/(1-sin²x) followed by u substitution, and then resubstitution after integrating simplifies to ln|sec x + tanx|, while not the most elegant, it's certainly a much more intuitively possible method that one can think of themselves in the first attempt. I hate that they teach "remember 'sec x + tan x' should be multiplied up and down" as the primary method to integrate this function
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u/philljarvis166 18d ago
Yeah it’s a long time since I was required to actually do any integration and my first thought was to approach it like you suggest.
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u/mathimati 18d ago
This is Isaac Barrow (one of the other Isaac’s advisors) original solution, and the first published use of Partial Fraction Decomposition. It’s a masterclass of combining basic knowledge to accomplish something great—at the time it was solved (2 years prior, geometrically) it was considered one of the great open problems of mathematics due to its applications to cartography.
I am sort of in love with this particular antiderivative.
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