An Innocent-Looking Integral Problem, But...
Вставка
- Опубліковано 30 тра 2024
- 0:00 Introduction - Setting Up the Integral
1:28 Integration Method 1 - Trigonometric Substitution
6:34 Integration Method 2 - Hyperbolic Substitution
11:08 Integration Method 3 - Euler Substitution
Today I share you one of the questions of the Calculus 1 quiz that I took when I was in freshman year in college. The problem looks so simple, but the integration required in the question actually takes skills and quite challenging. It is still a very important integral, and I will show you three methods to carry out the integration.
#integral #calculus #IntegrationBySubstitution #IntegrationByParts #LengthOfACurve
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nice video!
Thank you! I appreciate your comment.
Good 💯
ikr it is indeed a good problem (although students who have to solve the problem would feel differently).
The integral is far more straightforward if you substitute `x = sh t` as it is not hard to integrate (ch t)^2
That's what I did in the 2nd method.
Chebyshev also works
I'm afraid I couldn't quite get it. Care to elaborate?
@@CornerstonesOfMath might be spelling the dudes name wrong 🤣 but anyways for integrals of the form:
X^m (a+bx^n)^(p/q)
p/q whole num (Case 0):
expand the binomial
(m+1)/n whole (Case 1):
u = (a+bx^n)^1/q
(m+1)/n + p/q whole (Case 2):
u = (b + ax^-n)^1/q
you only actually do substitutions in case 1 and 2. If none of the conditions are met you gotta do something else but it seems to work whenever it’s in this form. Perhaps I haven’t come across enough problems yet tho
@@darcash1738 I see. Perhaps the name of the method is not popular, because the integral itself does not seem completely new to me.
However, I doubt whether it will lead to the new method. You can think of the function as x(1+x^-2)^(1/2), which fits Case 1, or just think of it as Case 2 with m = 0, n = -2. Both give the substitution method u = (1+x^2)^(1/2), but this only leads to
integral(1 to √2) (u^2)/(u^2-1)^(1/2) du.
If you let u = secθ to remove radical, it leads to integral of sec^3(θ).
The integral should become pretty intense if you do it right. x(1+x^-2)^(1/2) is case 1 like you said since (1+1)/-2 is whole or you could have it be x^0, and n = 2 for case 2. Case one means the substitution should be u = (a + bx^n) ^ (1/q), or
(1 + x^-2)^(1/2)
The way I remember it is thinking of a sort of matrix kinda.
(a b
b a) ^ 1/q
From there you basically have it. ab is in order so it is like in the original integral, but ba is out of order, so it is ax^-n. This is the easiest way to think of it imo which is why I’ve never forgotten it