Let integral be I
You have made a mistake when subbing dx back into the integralLet integral be I
let x^2 = tanθ
2x.dx = sec^2 θ .dθ
When x= 0, θ = 0
When x = INFTY, θ = pi/2
therefore, int(θ=0 to θ=pi/2) = 0.5 dθ
therefore, I = pi/4?
You can't use that substitution because you can't get rid off the x after you sub dx back.Let integral be I
let x^2 = tanθ
2x.dx = sec^2 θ .dθ
When x= 0, θ = 0
When x = INFTY, θ = pi/2
therefore, int(θ=0 to θ=pi/2) = 0.5 dθ
therefore, I = pi/4?
This can be so easily done using abstract algebra but I don't think that is HSC level.
The final result is nice though. Allows you to evaluate a sine Fresnel Integral without using complex analysis (you need Gamma function though).Hint: fugly completion of squares
Finding roots of unity then expressing in quadratic factors then applying partial fractions, then manipulating to get it into a suitable integral form:
Good job!Finding roots of unity then expressing in quadratic factors then applying partial fractions, then manipulating to get it into a suitable integral form:
By integrating both sides, we yield:
First and second fractions are logarithms, both elements inside the logarithm functions are monic quadratics, at infinity they cancel to 1, making the first 2 terms together 0, at x=0, the same happens and we get the logarithm of 1 which is again zero.
The second 2 terms are in the form for the atan function. After completing the square and manipulations, we evaluate them at infinity keeping in mind that at infinity atan is pi/2.
In the end we should arrive at
Also it would be awesome if people started posting these integrals in the integration marathon please =)
ahh yes lol, thanksYou have made a mistake when subbing dx back into the integral
ah yes i see thanksYou can't use that substitution because you can't get rid off the x after you sub dx back.
The only substitution you can use is x=tanx; which doesn't really help.
No heavy machinery required here, the simplest proof of this fact that I can think of is essentially identical to the standard proof of a course theorem.This can be so easily done using abstract algebra but I don't think that is HSC level.
I know haha, but proving those assumptions is a little beyond MX2, so I let people assume it.Lol rather large assumptions there Realise.
Plus Euler assumed them, why can't weLol rather large assumptions there Realise.
Do you do this questions using the conjugate root theorem?