HSC 2016 MX2 Integration Marathon (archive) (2 Viewers)

Drsoccerball · Jan 9, 2016

Re: MX2 2016 Integration Marathon

New question:

$\int (sinx+cosx)^n dx$

InteGrand · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Drsoccerball said:
New question:

$\int (sinx+cosx)^n dx$

$\noindent One way is to write $\cos x+ \sin x \equiv \sqrt{2}\cos \left(x -\frac{\pi}{4}\right)$ (or another equivalent sinusoid) and then use a reduction formula. I'll leave the integral for someone else.$

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
$\noindent I think Paradoxica meant that the $t$-substitution ends up making things nicer than what they started with (which isn't always the case).$

you mean basically never.

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Drsoccerball said:
$$ First take note that: $ \int_0^a{f(x)}dx=\int_0^a{f(a-x)}dx$

$I= \int_0^{\frac{\pi}{2}}{\frac{cos^ax}{sin^ax+cos^ax}}dx$

$I= \int_0^{\frac{\pi}{2}}{\frac{sin^ax}{sin^ax+cos^ax}}dx$

$2I= \int_0^{\frac{\pi}{2}}{\frac{cos^ax + sin^ax}{sin^ax+cos^ax}}dx$

$I= \frac{\pi}{4}$

should use something else other than "a", might confuse the newbies.

Drsoccerball · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
should use something else other than "a", might confuse the newbies.

Fixed.

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
$\noindent One way is to write $\cos x+ \sin x \equiv \sqrt{2}\cos \left(x -\frac{\pi}{4}\right)$ (or another equivalent sinusoid) and then use a reduction formula. I'll leave the integral for someone else.$

Except that doesn't give a specific form for which you can just plug in "n" and get the result. The solution to the integral can only be given in terms of the hypergeometric function.

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Drsoccerball said:
Fixed.

You're funny.

InteGrand · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
Except that doesn't give a specific form for which you can just plug in "n" and get the result. The solution to the integral can only be given in terms of the hypergeometric function.

This is why it'd probably be nicer as a definite integral with limits chosen so as to make terms vanish after IBP.

InteGrand · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
you mean basically never.

Actually, it is often the case for HSC integrals. Rarely though for the harder ones (which equally rarely appear in the HSC it seems!).

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
This is why it'd probably be nicer as a definite integral with limits chosen so as to make terms vanish after IBP.

I can't find any "nice" limits of integration.

InteGrand · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
I can't find any "nice" limits of integration.

$\noindent What about limits like $\frac{\pi}{4}$ and $\frac{\pi}{4}+\frac{\pi}{2}$?$

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
$\noindent What about limits like $\frac{\pi}{4}$ and $\frac{\pi}{4}+\frac{\pi}{2}$?$

Requires a quotient of two Gammas to express, Doesn't count in my eyes.

Drsoccerball · Jan 9, 2016

Re: MX2 2016 Integration Marathon

New Question

$\int{\frac{Lsinx+mcosx}{(asinx+bcosx)^2}}dx$

I think we should leave this one for the less experienced members.

Paradoxica · Jan 9, 2016

Re: MX2 2016 Integration Marathon

Drsoccerball said:
New Question

$\int{\frac{Lsinx+mcosx}{(asinx+bcosx)^2}}dx$

I think we should leave this one for the less experienced members.

$\int \frac{c \sin{x}+d \cos{x}}{(a \sin{x}+ b \cos{x})^2}dx$

FTFY

Always put a backslash before a function, and enclose the argument in braces.

Drsoccerball · Jan 10, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
$\int \frac{c \sin{x}+d \cos{x}}{(a \sin{x}+ b \cos{x})^2}dx$

FTFY

Always put a backslash before a function, and enclose the argument in braces.

ceebs for that...

Paradoxica · Jan 10, 2016

Re: MX2 2016 Integration Marathon

$\noindent \mathbf{Question for everyone else}$

$\int \frac{\text{d}x}{x \prod_{r=1}^{n}{\log_{(r)}{x}}}$ where $\log_{(k)}{x}$ denotes the $k$-nested logarithm.$$

InteGrand · Jan 10, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
$\noindent \mathbf{Question for everyone else}$

$\int \frac{\text{d}x}{\prod_{r=1}^{n}{\log_{r}{x}}}$ where $\log_{k}{x}$ denotes the $k$-nested logarithm.$$

$\noindent Should there be an $x$ on the denominator? Otherwise, there's no elementary solution I believe. Also, it'd probably be better to put the nesting index as a superscript, since subscripts already have a meaning for logs (log base). Doesn't matter much though since you specified what you meant by it.$

Paradoxica · Jan 10, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
$\noindent Should there be an $x$ on the denominator? Otherwise, there's no elementary solution I believe. Also, it'd probably be better to put the nesting index as a superscript, since subscripts already have a meaning for logs (log base). Doesn't matter much though since you specified what you meant by it.$

I only use the superscripted logarithm to indicate exponentiation of a logarithm.

InteGrand · Jan 10, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
I only use the superscripted logarithm to indicate exponentiation of a logarithm.

$\noindent What do you mean? You mean for something like this?: $\mathrm{e}^{\ln x}$.$

$\noindent Edit: oh right, you mean raising logs to powers.$

Drsoccerball · Jan 10, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
$\noindent \mathbf{Question for everyone else}$

$\int \frac{\text{d}x}{x \prod_{r=1}^{n}{\log_{(r)}{x}}}$ where $\log_{(k)}{x}$ denotes the $k$-nested logarithm.$$

Not sure if its this easy?

$I= \int{\frac{1}{x(\log{x}) J}}dx $ Where $ J = \frac{1}{\log 1} +\frac{1}{\log 2}+...+ \frac{1}{\log n}$

$I= \frac{1}{J \log{e}} \int{\frac{dx}{x \ln x}}$

$I=\frac{1}{J \log{e}} \ln (\ln{x} )) + C$

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