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Re: MX2 2015 Integration Marathon $ For $n\,>\,1$, evaluate $\int_{0}^{\infty} \frac{\text{ d}x}{(x+\sqrt{1+x^2})^n}$.

Re: MX2 2015 Integration Marathon ^ I think using the direct substitution u^2 = \tan(x) would have made the first part a bit easier (no need for a second substitution). Rest looks fine though :)

$ Let $4^a = x$, then \\ $ x+3x = 8$ \Rightarrow\, $4x = 8$ \Rightarrow\, $x = 2$. Thus, $4^a = 2$ \Rightarrow\, $2^{2a} = 2$, so $2a = 1$ \Rightarrow\, $a = \frac{1}{2}.$ $

Re: MX2 2015 Integration Marathon If I remember correctly, this q was also asked in the 2001 MX2 HSC Paper Q8 ...

I don't know of any particular software programs or the like but you could try some videos on YouTube/Khan Academy which utilise visual diagrams to explain how the volume of the curve rotated around axes is set out (for both slicing and shell methods).

Re: MX2 2015 Integration Marathon EASIER METHOD: $ Use the substitution $u = \frac{1}{x}$ to get $\int_{0}^{1} \frac{\text{ d}u}{\sqrt{2-(u-1)^2}}$ which becomes easier to evaluate. $

Re: MX2 2015 Integration Marathon BUMP :)

Yes.

Basically it means you can do one course or the other, but not both. So since you do MATH1151 you won't be able to do ECON1202 and vice versa.

Yes.

From the top of my head, for (i) I can think of \frac{x}{x^2+1} and for (ii) I can think of \frac{-1}{1+x^2} . For (iii), you will need to consider the case where the you split the interval into (-\infty, M_1), [M_1, M_2] and (M_2, \infty) and apply the Extreme Value Theorem + definition...

Re: MX2 2015 Integration Marathon NEW QUESTION: $ Find $\int_{1}^{\infty} \frac{\text{ d}x}{x{\sqrt{x^2+2x-1}}}$ $

Vowels will come into play if the question actually specified something about it, in this case it is just general for any set of 4 letters you pick from the original set.

Not about vowels, but about taking repetitions of letters. We have WEDNESDAYS, so we can say that the total number of distinct letters is just W, E, D, N, S, A, Y. There are 7 elements here, so if we want to pick 4, there will be 7C4 total possibilities (this ensures no letter will be repeated...

So it should be 7C4 + (3C1 X 6C2) + 3C2 = 83? This is just referring to the different selections (combinations) only.

Don't know if my explanation is clear cut, you will need read the textbook so you can follow more detailed explanations as to knowing when to use what. Cambridge seems to explain these things quite well, so have a look at that if you still don't understand :)

NOTE that nPr and nCr are just definitions that you need to follow, with the relation that nPr = r! X nCr, since you arranging each of the r elements in your chosen subset to obtain all ORDERED sets with the same number of r elements in them. Elements can be different i.e. {A, B}, {A, C}, {B, C}...

Factorials are used when considering order of objects so if I have a set {A, B, C} then the total number arrangements are 3 X 2 X 1 = 3! = 6. Here, repetition is not allowed. With combinations, you divide the total number of permutations by the total number of arrangements for the elements being...

Sorry, it should be C, I misread the question. In combinations, the order in which the elements are arranged is disregarded, unlike in permutations where the order matters.

Essentially you want to solve \frac{^{2n+2}C_n}{^{2n-2}C_n} = \frac{99}{7}. Use the definition of ^{n}C_{r} to find your value of n.