HSC 2018-2019 MX2 Integration Marathon (1 Viewer)

stupid_girl

Member
Re: HSC 2018 MX2 Integration Marathon

This is a skeleton solution.
By substituting u=(x-2)/sqrt(2) and considering f(x)+f(-x), the integral can be re-written as
$\bg_white \frac{\sqrt{2}}{8}\int_0^{\sqrt{2}-1}\frac{\sqrt{1+u^4}}{1-u^4}du$

A tangent substitution will turn it into a format that Wolfram can solve...finally
$\bg_white \frac{\sqrt{2}}{8}\int_0^{\frac{\pi}{8}} \frac{\sqrt{1+\tan^4\theta}}{1-\tan^2\ \theta}d\theta$
https://www.wolframalpha.com/input/?i=integrate+sqrt(1+tan^4+x)+/+(1-tan^2+x)

I know Wolfram used hyperbolic tangent substitution but it is also solvable in MX2 by secant substitution.

Alternatively, if you don't mind handling improper integral, you can do some algebraic manipulation to get:
$\bg_white \frac{\sqrt{2}}{16}\int_0^{\sqrt{2}-1}\frac{u^{-2}-1}{\left(u^{-1}+u\right)\sqrt{\left(u^{-1}+u\right)^2-2}}du+\frac{\sqrt{2}}{16}\int_0^{\sqrt{2}-1}\frac{u^{-2}+1}{\left(u^{-1}-u\right)\sqrt{\left(u^{-1}-u\right)^2+2}}du$
Substituting v=u-1+u and w=u-1-u will lead to two improper (but solvable) integrals because u-1 blows up at 0.
Not sure if anyone attempted to go further from this.

If you are careful with the manipulation, you should have got the final answer.
$\bg_white \int_{4-\sqrt{2}}^{\sqrt{2}}\frac{\sqrt{(x^2-6x+10)(x^2-2x+2)}}{(x^2-4x+2)(x^2-4x+6)(4+2^x)}dx$
$\bg_white =\frac{1}{4}\int_{4-\sqrt{2}}^{\sqrt{2}}\frac{\sqrt{\left(x-2\right)^4+4}}{\left(\left(x-2\right)^4-4\right)(1+2^{x-2})}dx$
$\bg_white =\frac{1}{4}\int_{2-\sqrt{2}}^{\sqrt{2}-2}\frac{\sqrt{y^4+4}}{\left(y^4-4\right)(1+2^y)}dy$
$\bg_white =\frac{1}{4}\int_0^{\sqrt{2}-2}\frac{\sqrt{y^4+4}}{\left(y^4-4\right)}dy$
$\bg_white =\frac{\sqrt{2}}{8}\int_0^{\sqrt{2}-1}\frac{\sqrt{1+u^4}}{1-u^4}du$
$\bg_white =\frac{\sqrt{2}}{8}\int_0^{\frac{\pi}{8}} \frac{\sqrt{1+\tan^4\theta}}{1-\tan^2\ \theta}d\theta$
$\bg_white =\frac{1}{8}\int_0^{\frac{\pi}{8}}\frac{\cos (2\theta)}{\sqrt{2-\sin^2 (2\theta)}}d\theta+\frac{1}{8}\int_0^{\frac{\pi}{8}}\frac{\sec^2 (2\theta)}{\sqrt{2+\tan^2 (2\theta)}}d\theta$
$\bg_white =\left[\frac{1}{16}\sin^{-1}\left(\frac{\sin2\theta}{\sqrt{2}}\right)+\frac{1}{16}\ln\left(\tan2\theta+\sqrt{2+\tan^2 2\theta}\right)\right]_0^{\frac{\pi}{8}}$
$\bg_white =\frac{1}{16}\left(\frac{\pi}{6}\right) + \frac{\ln\left(1+\sqrt{3}\right)}{16}-\frac{\ln\sqrt{2}}{16}$
$\bg_white =\frac{\pi}{96}+\frac{\ln\left(1+\sqrt{3}\right)}{16}-\frac{\ln2}{32}$

stupid_girl

Member
Re: HSC 2018 MX2 Integration Marathon

I saw another approach on the internet...however the back substitution may be slightly messier.

stupid_girl

Member
Re: HSC 2018 MX2 Integration Marathon

This is slightly tedious.
$\bg_white \int_2^4\left(\sqrt{x} + \sqrt[5]{\frac{x^{\frac{76x+75}{4x}} - \left(\log_3 x\right)^{19}}{x^{\frac{72x-73}{3x}} + \left(\log_5 x\right)^{24}}} + \frac{1}{\sqrt{x}}\right)\left(\sqrt{x} - \sqrt[5]{\frac{x^{\frac{76x+75}{4x}} - \left(\log_3 x\right)^{19}}{x^{\frac{72x-73}{3x}} + \left(\log_5 x\right)^{24}}} + \frac{1}{\sqrt{x}}\right) \log_2\left(\frac{x}{e}\right) dx$

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-insert title here-
Re: HSC 2018 MX2 Integration Marathon

I saw another approach on the internet...however the back substitution may be slightly messier.
This only works for x>0 because the resultant primitive does not have a derivative at 0, yet the function to be integrated is clearly defined at 0.

stupid_girl

Member
#83 and #88 are still outstanding and this is a new one.
Feel free to share your attempt.
$\bg_white \int_0^1\left(\sqrt{4-4^x}\sqrt{7\left(16^x\right)+16^{2x}-5\left(4^{3x}\right)}+\sqrt[3]{2^{10x}-2^{8x+1}}\right)dx$

stupid_girl

Member
Re: HSC 2018 MX2 Integration Marathon

This one requires the same trick you've seen.
$\bg_white \int_{-\frac{3}{4}}^{\frac{4}{3}}\frac{1}{(1+2\cos^2(\pi x))}dx$
The answer for this one is
$\bg_white \frac{25\sqrt{3}}{36}$

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stupid_girl

Member
Re: HSC 2018 MX2 Integration Marathon

This is slightly tedious.
$\bg_white \int_2^4\left(\sqrt{x} + \sqrt[5]{\frac{x^{\frac{76x+75}{4x}} - \left(\log_3 x\right)^{19}}{x^{\frac{72x-73}{3x}} + \left(\log_5 x\right)^{24}}} + \frac{1}{\sqrt{x}}\right)\left(\sqrt{x} - \sqrt[5]{\frac{x^{\frac{76x+75}{4x}} - \left(\log_3 x\right)^{19}}{x^{\frac{72x-73}{3x}} + \left(\log_5 x\right)^{24}}} + \frac{1}{\sqrt{x}}\right) \log_2\left(\frac{x}{e}\right) dx$
The answer for this one is
$\bg_white 25+\frac{3}{2}\ln2-\frac{17}{\ln2}$

stupid_girl

Member
#83 and #88 are still outstanding and this is a new one.
Feel free to share your attempt.
$\bg_white \int_0^1\left(\sqrt{4-4^x}\sqrt{7\left(16^x\right)+16^{2x}-5\left(4^{3x}\right)}+\sqrt[3]{2^{10x}-2^{8x+1}}\right)dx$
The answer for this one is
$\bg_white \frac{3}{\ln2}$

stupid_girl

Member
This is a new one. Feel free to share your attempt.
$\bg_white \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{\tan^4x+\cot^4x}{1+\tan^{\frac{2019}{2020}}x}dx$

fan96

617 pages
$\bg_white I = \int^{\pi/3}_{\pi/6} \frac{\tan^4x+\cot^4x}{1+\tan^{\frac{2019}{2020}}x} \,dx$

First we note that $\bg_white (\tan x)^{-1} = \cot x$ for $\bg_white x\in [\pi/3, \, \pi/6]$.

Using

$\bg_white \int_a^{b} f(x) \, dx =\int_a^{b} f(a+b-x) \, dx$,

$\bg_white \tan(\pi/2 - x) = \cot x$, and $\bg_white \cot(\pi/2 - x) = \tan x$

$\bg_white \therefore I = \int^{\pi/3}_{\pi/6} \frac{\tan^4x+\cot^4x}{1+\cot^{\frac{2019}{2020}}x} \,dx$

$\bg_white = \int^{\pi/3}_{\pi/6} \frac{\tan^4x+\cot^4x}{1+\frac 1 {\tan^{\frac{2019}{2020}}x}} \,dx$

$\bg_white = \int^{\pi/3}_{\pi/6} \tan^{\frac{2019}{2020}}x \cdot \frac{\tan^4x+\cot^4x}{1+\tan^{\frac{2019}{2020}}x} \,dx$

$\bg_white \implies 2I = \int^{\pi/3}_{\pi/6} \tan^4x+\cot^4x \,dx$

$\bg_white = \int^{\pi/3}_{\pi/6} \tan^2x\sec^2x-(\sec^2x-1)+\cot^2x\csc^2x -(\csc^2x-1) \,dx$

$\bg_white = \left[\frac 1 3 \tan^3x - \tan x - \frac 13 \cot^3 x + \cot x + 2x \right]^{\pi/3}_{\pi/6}$

$\bg_white =\frac \pi 3 + \frac{16\sqrt 3}{27}$

$\bg_white \iff I = \frac \pi 6 + \frac{8\sqrt 3}{27} \approx 1.0368$

stupid_girl

Member
A new one
$\bg_white \int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{x\sin x}{\left(1+\tan^2\frac{x}{2}\right)\left(x-\sin x\right)^3}dx$

The answer looks quite ugly and probably can't be simplified further. Taking common denominator and expanding out will make it really messy.
$\bg_white \frac{6+4\sqrt{2}}{\left(\pi-2\sqrt{2}\right)^2}-\frac{1}{\left(\pi-2\right)^2}$

stupid_girl

Member
A new one
$\bg_white \int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{x\sin x}{\left(1+\tan^2\frac{x}{2}\right)\left(x-\sin x\right)^3}dx$

The answer looks quite ugly and probably can't be simplified further. Taking common denominator and expanding out will make it really messy.
$\bg_white \frac{6+4\sqrt{2}}{\left(\pi-2\sqrt{2}\right)^2}-\frac{1}{\left(\pi-2\right)^2}$
As usual, reverse quotient rule problems are easy to set but difficult to solve. It becomes a piece of cake IF (a big if) you can spot it.
$\bg_white \int\frac{x\sin x}{\left(1+\tan^2\frac{x}{2}\right)\left(x-\sin x\right)^3}dx=-\frac{\left(1+\cos\ x\right)^2}{4\left(x-\sin x\right)^2}+c$

stupid_girl

Member
This is a new one.
$\bg_white \int_1^2\frac{\tan^{-1}\sqrt{2x-1}}{\left(2x\tan^{-1}\sqrt{2x-1}-\sqrt{2x-1}\right)^3}dx$

Once again, the answer looks quite ugly.
$\bg_white \frac{1}{\left(\pi-2\right)^2}-\frac{9}{4\left(4\pi-3\sqrt{3}\right)^2}$

stupid_girl

Member
This is a new one.
$\bg_white \int_1^2\frac{\tan^{-1}\sqrt{2x-1}}{\left(2x\tan^{-1}\sqrt{2x-1}-\sqrt{2x-1}\right)^3}dx$

Once again, the answer looks quite ugly.
$\bg_white \frac{1}{\left(\pi-2\right)^2}-\frac{9}{4\left(4\pi-3\sqrt{3}\right)^2}$
Hint: You may consider the following substitution.
$\bg_white u=2x\tan^{-1}\sqrt{2x-1}-\sqrt{2x-1}$

stupid_girl

Member
Using the above substitution, it should be obvious that
$\bg_white \int\left(\tan^{-1}\sqrt{2x-1}\right)\left(2x\tan^{-1}\sqrt{2x-1}-\sqrt{2x-1}\right)^kdx$
$\bg_white =\frac{\left(2x\tan^{-1}\left(\sqrt{2x-1}\right)-\sqrt{2x-1}\right)^{k+1}}{2\left(k+1\right)}+c for k\neq-1$
The definite integral can be evaluated easily.

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stupid_girl

Member
A few new integrals

If you can solve one of them, then you can probably solve all of them.
$\bg_white \int\left(x+\sqrt{x^2+1}\right)^{\sqrt{2019}}dx$
$\bg_white \int\left(\sqrt{x^2+1}-x\right)^{\pi}}dx$
$\bg_white \int\left(\sec x+\tan x\right)^{2019}\sec^2xdx$
$\bg_white \int\left(\cot x+\csc x\right)^{\frac{22}{7}}\csc^2xdx$

stupid_girl

Member
Why is it getting so quiet?

Let's have a new integral. Show that
$\bg_white \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\ln\left(1+\pi^x\right)}{4\cot x+5\csc x-\sin x}dx=\frac{9-4\sqrt{3}}{36}\ln\pi^{\pi}$

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stupid_girl

Member
One more...feel free to share your attempt.
$\bg_white \int_0^{\pi}\frac{\ln\left(1+\pi^{\cos x}\right)}{2\sec x-\cos x+2\tan x}dx=\left(4-\pi\right)\ln\sqrt{\pi}$

HeroWise

Active Member
After multiplying cosx and rearranging the quadratic im stuck a little hint please!

Edit:
DAMMIT im not getting the sqrt pi

GOT IT!

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HeroWise

Active Member
Use Kings property for another integral, add them, and reduce to a smaller fraction.
After that use weierstrauss and after simplifying use partial fractions. Then integrate normally. Sub in the values and u got it Nice question