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Re: HSC 2017 MX2 Integration Marathon It is purposefully obfuscated but I think average MX2 student should be able to interpret it. End of the day, <f,g>w and <f,g> are just two real numbers. The answer would be the same if I call them star and moon, and ask for star/moon. By the way, I...

Re: HSC 2017 MX2 Integration Marathon \text{Define }<f,g>=\int_0^1 f(s)g(s)ds\text{ and }<f,g>_w=\int_0^1 f(t)g(t)w(t)dt\text{. Find }\frac{<f,g>_w}{<f,g>}\text{ when}\\f(x)=x^{a-1},g(y)=(1-y)^{b-1}\text{ and }w(z)=(z+k)^{-a-b}\text{ for positive real constants a,b and k.}

It is not necessarily a pyramid. It may have different tilt/twist at different level. I think integration is the way to go.

Re: HSC 2017 MX2 Marathon When you are asked to prove a given reduction formula, the efficient way is usually by differentiation.

Re: HSC 2017 MX2 Integration Marathon Hope you are familiar with trig identities.:tongue: \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{tan^{-1}(2^{\sqrt{(sec x+1)(sec x-1)}}-1)}{(1-sin x)}dx

I don't think it works. This is not a GS. nX = n! + n (n-2)! + n (n-3)! + ......

Re: HSC 2017 MX2 Integration Marathon For Q4, if you stare at it long enough, you should be able to use reverse quotient rule. Let y=1+ln x, then y'=1/x The integrand can be written as (y-1)/y^2 =(x'y-xy')/y^2. Now, the answer should be obvious.

Re: HSC 2017 MX2 Integration Marathon Are you sure Q3 has closed form solution?

Re: HSC 2017 MX2 Marathon This geometry question doesn't involve any advanced knowledge but requires a bit of creativity.:tongue: ABCD is a quadrilateral with three equal sides AB,BC and CD. Show that the mid-point of AD lies on a circle with diameter BC if and only if the area of ABCD is a...

Re: HSC 4U Integration Marathon 2017 This one is a bit tedious. \int_{0}^{\frac{\pi^2}{16}} tan\sqrt{x}+\frac{2}{1+tan\sqrt{x}}

Re: HSC 4U Integration Marathon 2017 \int_{-\pi/2}^{\pi/2} \frac{(2+3\pi^x)(sin^{2016} x)+(4+5\pi^x)(cos^{2016} x)}{(1+\pi^x)(sin^{2016} x+cos^{2016} x)}dx

Re: HSC 4U Integration Marathon 2017 The approach is correct. Well done! Just a few minor things to point out. In (a)(i), f(G(x))g(x)>=f(x)g(x) because g(x)>=0 In (a)(ii), phi(x)>=0 for all 0<x<=1 In (b)(ii), the result of (a)(ii) can be used because 0<=1-g(t)<=1

Re: HSC 4U Integration Marathon 2017 This question relies on Fundamental Theorem of Calculus and substitution.:tongue:

Re: HSC 4U Integration Marathon 2017 That's correct!!! :lol: I am not aware of any more elegant way to do it.

Re: HSC 4U Integration Marathon 2017 Sorry it's my fault. I accidentally missed out the exponent on the numerator when I typed it in latex. Nevertheless, the mistyped integral is not messy. I just attempted it and the integrand can be broken into 0.5 sin x/(cos x)^7 and -0.5(sec x)^6...

Re: HSC 4U Integration Marathon 2017 This definite integral must be positive because the integrand is a complete square.:smile: (Sorry for the typo in numerator this afternoon.) \int_{0}^{\frac{\pi}{4}} \frac{(sin(2x+\frac{\pi}{4})-\frac{1}{\sqrt{2}})^{4}}{4cos^{8}x}

Re: HSC 4U Integration Marathon 2017 It's correct, but I wonder how Paradoxica did it by inspection.:headbang:

Re: HSC 4U Integration Marathon 2017 I hope this one isn't too bad. :tongue: \int_{0}^{\pi/4} \frac{log_{2}(cos^{sin 2x}x)}{\pi(3+cos4x)}

Why not? UNSW Handbook Australian School of Business Once again, please read 9.4.2 The minimum requirement for Science is 84UOC. If you use all the remaining UOC for Commerce, you have 108UOC. This is sufficient for you to have a double major in Commerce without studying past the normal length...

I would suggest MATH2871. It teaches you the computer software SAS and involves some basic programming techniques. I believe that it can be count as general education as well.