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Re: MX2 2016 Integration Marathon \int_0^{\pi}{\frac{\sin{(2015x) dx}}{\sin{x}}}

Re: HSC 2016 2U Marathon Nice. Now try the next question.

Re: HSC 2016 2U Marathon Yes I mistyped the question. Sorry about that :P but +2 makes the question nicer you can do the same thing for +1 as well.

Re: HSC 2016 2U Marathon $ Firstly it is important to note that the graph's inside the $ \log $ must be positive. $ x^2 -3x +2 >0 $ Equality occurs at $ x=1,2 6-x^2+4x>0 $ Equality occurs at $ x= -1.16, 5.16 $ Getting rid of the $ \log 's $ we get : $ 2x^2-7x-4 > 0 $...

Re: HSC 2016 2U Marathon Ill post my answer hold up.

Re: HSC 2016 2U Marathon $ Please past graduates do not help in this next problem. I don't want to waste an amazing question. $ $ Simplify : $ \frac{\log_{\frac{a}{b}}x \log_{\frac{a}{c}}{x}}{\log_a ^2 x} +\frac{\log_{\frac{b}{c}}x \log_{\frac{b}{a}}{x}}{\log_b^2 x}...

Re: HSC 2016 2U Marathon Those are not the answers.

Re: HSC 2016 3U Marathon Try using the fact that adding, multiplying, dividing anything etc... will remove the equality of the inequality if the operation isn't done on both sides.

Re: MX2 2016 Integration Marathon So there's 2 cases one for a=0 and one for all other values ?

Re: MX2 2016 Integration Marathon After a couple of manipulations we get the integral I= \frac{1}{a}\ln{|x|} + \frac{2x}{a^2} -\frac{2}{a} \int{\frac{\sqrt{x+a}}{a\sqrt{x}}}dx $ The remaining integral can be done with a substitution $ x= a \tan ^2 \theta $ Which give us the integral of $...

Re: MX2 2016 Integration Marathon 2 substitution leave the integral in the form of \int{\sec{x}}dx In a rush lel jokes beaten

Re: MX2 2016 Integration Marathon You got to teach me :P (no joke pls teach me)

Re: HSC 2016 3U Marathon $ Use the graph $ f(x)= \frac{x^e}{e^x} $ to prove that : $ e^{\pi}>\pi ^e

Proof of results.

Re: MX2 2016 Integration Marathon I think my method would be faster if I knew inverse hyperbolic functions.

Re: MX2 2016 Integration Marathon I havn't done p.f like that before.

Re: MX2 2016 Integration Marathon I've never done it before but it came in my mind for this question.

Re: MX2 2016 Integration Marathon LOL I =\frac{-du}{(u+1)\sqrt{u^2+1}} = \frac{1}{2}\int ({\frac{u-1}{\sqrt{u^2+1}} - \frac{\sqrt{u^2+1}}{u+1}})du Yeah don't ask...

Re: MX2 2016 Integration Marathon Hmmm I shouldnt have used partial fractions it made a non elementary integral.

What's the question?