Comment by Captain Chicky on Proving...
two things - 1. why do you want to use complex analysis? real methods will be orders of magnitude less work and easier. 2. use a rectangular contour with an indent and take the limit as that indent...
View ArticleComment by Captain Chicky on Proving...
@KamalSaleh it’s fine :) their solution is interesting enough but still leaves with an integral that you need to use real methods to evaluate kek tho
View ArticleComment by Captain Chicky on Proving...
Damn this is actually very smart. How did you think of using a linear term on the numerator? (My complex analysis class didn’t show any strategy of integration like this 😭)
View ArticleComment by Captain Chicky on Are there any good trig/precalc textbooks to...
Honestly just grind past problems that’s what I always did lol and I made AIME 3 times
View ArticleComment by Captain Chicky on Volume of a solid with an elliptical base using...
Can you draw a diagram on what you mean, I might be misinterpreting this, (is this calculus)?
View ArticleComment by Captain Chicky on Prove: $\int_1^3\sqrt{x^3}e^xdx \in (2e^3,...
Isn’t an open interval weaker than a closed one? Your proof shouldn’t have a problem because in solving for the closed interval bound it automatically includes the open interval bound no?
View ArticleComment by Captain Chicky on Evaluating a challenging definite integral
@Cognoscenti he converted it to a contour integral about the unit circle, and then applied the residue theorem. (Have you learned complex analysis yet? If not then rip)
View ArticleComment by Captain Chicky on Work of a vector field along a curve
seems like your initial vector field was a typo then
View ArticleComment by Captain Chicky on How to integrate $A/(x^B + C)$?
wolframalpha.com/input?i=integrate+1%2F%28x%5Eb%2Bc%29dx It has a hypergeometric series as an antiderivative if you want to work with that lol
View ArticleComment by Captain Chicky on Constant Acceleration
In classical mechanics, you can use the Galilean transform to transform between any inertial observer, but the answers you get will always be consistent
View ArticleComment by Captain Chicky on What non-trivial continuous functions $f$ such...
Let $a=-\infty$, then $f(x)=\operatorname{sinc}^k(x)$ works for $k$ from 1 to 6 (math.stackexchange.com/questions/170747/…)
View ArticleAnswer by Captain Chicky for Can the second integral of $x^x$ be expressed in...
This is probably not a satisfactory answer, and possibly an abuse of notation. If this does not work, just downvote it and I will delete later.As per...
View ArticleClik here to view.
Answer by Captain Chicky for Solution to $2^x=x!$
I cannot see any form of analytical solution possible to this equation. Instead, you can set a function $$f(x)=2^x-\Gamma(x+1)$$ and perform numerical methods for the second solution. You can use...
View ArticleClik here to view.
Answer by Captain Chicky for Need help with finding the correct region for...
Your region is the inner parallelogram.(You can see this clearly if you break each of the two conditions into two separate inequalities.)Btw when evaluating this integral you might want to try a change...
View ArticleAnswer by Captain Chicky for Using L'Hospital rule, how do we tell that...
When you plug in $0$ for the limit you’re plugging in $0$ for $x$ not $t$.$$\lim_{x \to 0+} \frac{\int_0^x \sin^{-1} t^2 dt}{\sin(x^2)}$$ basically means $$\frac{\int_0^0\sin^{-1} t^2...
View ArticleAnswer by Captain Chicky for What happens when you find the derivative of...
We can write the absolute value as $\sqrt{(x^2-1)^2}$, which we can differentiate with the chain...
View ArticleClik here to view.
Answer by Captain Chicky for Evaluation of...
The region in question is a slanted ellipse. Using change of variables for ellipses we have$$u=x-\frac{y}{2}\\v=\frac{\sqrt 3y}{2}$$Our region is thus mapped into the unit circle $u^2+v^2=1$ with...
View ArticleAnswer by Captain Chicky for Upper bound on |zf(z)| in a complex keyhole...
Okay, so regarding the circular paths in keyhole contours, I usually parameterize the contours and use various inequalities to prove the integrals on them go to $0$. In your case, here is an example.We...
View ArticleAnswer by Captain Chicky for Is there a way of solving this equation?
Raising both sides to the power of $e$ gives$$x=e^{x-3}$$Now recall the Lambert W function, or product log function, which has the property that$$W\left(xe^x\right)=x$$For positive $x$ (Which is...
View ArticleAnswer by Captain Chicky for What is $\int e^{e^x+x}dx$
I believe it would be much more clear to write our all of your variables and differentials in terms of the new substitution variable.Recall that for some substitution $u=f(x)$, we can see...
View Article
