HSC 2017 MX2 Integration Marathon (archive) (3 Viewers)

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

leehuan said:
$\int_0^\pi t \prod_{\beta=1}^\infty \cos\left(\frac{t}{2^\beta}\right)dt$

So is the answer 2?

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
So is the answer 2?

$\noindent The integrand is $\sin t$, so yes, answer is $2$.$

si2136 · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
Which methods are you talking about? Inspection in general should be fine (might depend on wording of question), and you could differentiate your answer for indefinite integrals if you wanted to show 'working'.

Particularly this method: http://www.nabla.hr/Z_MemoHU-128.htm

The Ostrogradsky-Hermite Method

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

Paradoxica said:
That doesn't work, it's still a non-elementary integral.

The answer returned by Mathematica is

xlog(1+e^x) + DiLog(-e^x) +C

The DiLogarithm is non-elementary and can only be simplified under specific circumstances, none of which are met by this integral.

What do you mean by non-elementary? You can still integrate it?

Paradoxica · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
What do you mean by non-elementary? You can still integrate it?

Depends on what you accept as an answer. The HSC definitely doesn't have the DiLogarithm in it, so such a question would never appear.

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
Here's a much easier one.

Find $\int \sqrt{x^{2} + 4x + 8}\,\mathrm{d}x$ .

$\text{Is the answer}\hspace{0.5cm}\frac{(x+2)\sqrt{x^{2}+4x+8}}{2}+2\ln|\frac{\sqrt{x^{2}+4x+8}+x+2}{2}|+\text{c ?}$

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

Paradoxica said:
Depends on what you accept as an answer. The HSC definitely doesn't have the DiLogarithm in it, so such a question would never appear.

What's a DiLogarithm? Is it just like a log or ln thing raised to a power? i.e. $(\ln|1+e^{x}|)^{2}$

omegadot · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
What do you mean by non-elementary? You can still integrate it?

$$\noindent When we say an integral cannot be integrated in \textit{elementary terms} it means no primitive for the function can be found in terms of the so-called elementary functions. The elementary functions are polynomials, roots, exponential, logarithmic, the trigonometric functions and their inverses, and compositions and combinations of these functions. Functions which are not elementary are known as \textit{special functions}. There are many, many special functions. They have names like the Bessel function, the gamma function, the dilogarithmic function, and so on. At HCS level you only ever meet with the elementary functions, unless it is a Question 16 type question with the function in disguise.$

omegadot · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
What's a DiLogarithm? Is it just like a log or ln thing raised to a power? i.e. $(\ln|1+e^{x}|)^{2}$

$$\noindent Ask yourself the question what is the integral $\int \frac{dt}{t}?$

$\noindent Of course you will immediately say it is the natural logarithm! Well this is just a convenient way of papering over the fact that until we expanded our function set to `include' the logarithmic function we had not way of providing an answer to such a question. In fact, the natural logarithmic function is simply defined as$

$\ln x = \int^x_1 \frac{dt}{t}.$

$$\noindent This definition has all the normal properties we expect for the logarithm. For example, using this definition it can be shown that $\ln (xy) = \ln x + \ln y$. Try it! More importantly however is the logarithmic function proves to be incredibly useful - it just seems to pop up all over the place.$

$\noindent The dilogarithm on the other hand is an example of a \textit{special function}. Sure, it pops up every now and again, but no where near as many times as the logarithmic function. In fact, the dilogarithmic function can be defined by$

$\text{Li}_2 (x)= -\int^x_0 \frac{\ln (1 - t)}{t} \, dt.$

$\noindent When you try to find this integral using those methods you are familiar with you will quickly discover it just is not possible to integrate the damn thing. And in fact it cannot be done in terms of elementary functions.$

$\noindent Of course, knowing whether this can be done in advanced would be a useful thing to know and can be decided using an algorithm known as the \textit{Risch Algorithm}, but to explain what this is would take us too far afield. Finally, you may be wondering why the name `dilogarithm'. Well, it truns out the dilogarithm is one way to generalise the ordinary natural logarithm (there are other ways), but again to understand why this is the case depends on a knowledge of Taylor series.$

BenHowe · Feb 25, 2017

Re: HSC 2017 MX2 Integration Marathon

omegadot said:
$\noindent You might like to try out the rule on the following question. And just for fun, let us find this integral in three different ways with a few hints provided along the way.$

$\noindent Find $\int \frac{x^2}{(x \sin x + \cos x)^2} \, dx $ as follows$

$\noindent \textsc{Method I}: Use the reverse quotient rule method.$

$$\noindent \textsc{Method II}: Use a substitution of $x = \tan u$ following by a substitution of $t = \tan u - u$

$\noindent \textsc{Method III}: By first observing that $\frac{d}{dx} (x \sin x + \cos x) = x \cos x$, use integration by parts.$

I'm not sure how I'm suppose to find the u term. Am struggling -_-

BenHowe · Feb 25, 2017

Re: HSC 2017 MX2 Integration Marathon

omegadot said:
$\noindent You might like to try out the rule on the following question. And just for fun, let us find this integral in three different ways with a few hints provided along the way.$

$\noindent Find $\int \frac{x^2}{(x \sin x + \cos x)^2} \, dx $ as follows$

$\noindent \textsc{Method I}: Use the reverse quotient rule method.$

$$\noindent \textsc{Method II}: Use a substitution of $x = \tan u$ following by a substitution of $t = \tan u - u$

$\noindent \textsc{Method III}: By first observing that $\frac{d}{dx} (x \sin x + \cos x) = x \cos x$, use integration by parts.$

$v=xsinx+cosx,\frac{dv}{dx}=xcosx\\\text{So}\hspace{0.1cm}u=sinx-xcosx\hspace{0.1cm}\text{This is because the power of x must alternate to use}\\sin^{2}x+cos^{2}x=1\hspace{0.1cm}\text{and negative sign to account for minus in the quotient rule to use this result. It works out}\\\text{Does anyone have any tips for 'seeing' this stuff? It took me a while and a lot of error...}$

Drsoccerball · Feb 26, 2017

Re: HSC 2017 MX2 Integration Marathon

Easy one : $\int{\frac{\sin{x}}{\sin{x} + \cos{x}}}$

jathu123 · Feb 26, 2017

Re: HSC 2017 MX2 Integration Marathon

Drsoccerball said:
Easy one : $\int{\frac{\sin{x}}{\sin{x} + \cos{x}}}$

$$Observe that $\sin x \equiv \sin x + \cos x +(\sin x - \cos x) -\sin x \\ \therefore I = \int \frac{\sin x + \cos x}{\sin x + \cos x}$ d$x+\int \frac{\sin x - \cos x}{\sin x + \cos x}$ d$x -\int \frac{\sin x}{\sin x + \cos x}$ d$x \\ \\ 2I=x-\ln\left | \sin x + \cos x \right | \\ \Rightarrow I=\frac{1}{2}\left ( x-\ln\left | \sin x + \cos x \right | \right )+C$

nancy99 · Feb 27, 2017

Re: HSC 2017 MX2 Integration Marathon

frog1944 · Feb 27, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
That's why I'm planning to use only known methods in my HSC externals. I asked the head teacher and I'm allowed to use any method as long as they have formal documentation on it and it leads to the answer (For ex. Heaviside Cover Up)

I'm sorry, but I'm slightly confused. For the HSC are we allowed to use methods other than that explicitly taught in the HSC for integration? Does this apply to other topics in math? Or is this a special case that your school is allowing?

si2136 · Mar 3, 2017

Re: HSC 2017 MX2 Integration Marathon

Are there any faster methods on integrating even powers of trig functions with double angle? Cheers

Paradoxica · Mar 3, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
Are there any faster methods on integrating even powers of trig functions with double angle? Cheers

Extracurricular Methods, sure.....

Are we talking with bounds or without bounds?

si2136 · Mar 3, 2017

Re: HSC 2017 MX2 Integration Marathon

Paradoxica said:
Extracurricular Methods, sure.....

Are we talking with bounds or without bounds?

With bounds, thanks

Paradoxica · Mar 3, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
With bounds, thanks

I still can't interpret your question. Perhaps provide some specific classes of functions.

si2136 · Mar 3, 2017

Re: HSC 2017 MX2 Integration Marathon

Paradoxica said:
I still can't interpret your question. Perhaps provide some specific classes of functions.

Do you mean examples?

What I meant was like I = Int(sec^6) dx or I = Int(tan^8) dx

A trig function to the power of a high even number, which using half angle would be tedious and time-costly.

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