HSC 2018-2019 MX2 Integration Marathon (1 Viewer)

HeroWise

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Re: HSC 2018 MX2 Integration Marathon

Lets campaign the teachers to give mrbunton the mark!
 

fan96

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Re: HSC 2018 MX2 Integration Marathon

In the case of a misprinted question, I think the best thing to do here would just be to give everyone the marks.
 

jyu

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Re: HSC 2018 MX2 Integration Marathon

Surely it's been written wrong? Only way I can think of using Taylor, but that is outside 4U scope.
Try replacing both e^x with e^(x^2).
 

Paradoxica

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Re: HSC 2018 MX2 Integration Marathon

The above question is literally impossible, in the sense there is no elementary closed form in terms of a finite combination of elementary operations and HSC-level constants.
 

stupid_girl

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Re: HSC 2018 MX2 Integration Marathon

Show that


 
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fan96

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Re: HSC 2018 MX2 Integration Marathon



Using the identities:





the integral reduces to:





Perform a substitution to get



The substitution would probably have given a quicker answer, but the first thing that came to mind was another trig sub:



Giving:



Noting that for all real and so for the bounds of this integral, we may simplify to get





Note: and can be evaluated by using an appropriate right angled triangle.





Expanding the square and rationalising the denominator gives

 
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Paradoxica

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Re: HSC 2018 MX2 Integration Marathon

Continuing on from the simplification



Reverse the chain rule twice to obtain:



Complete the substitution to obtain:



Using the former table of standard integrals, the integral evaluates to:



Which "simplifies" to:

 
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fan96

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Re: HSC 2018 MX2 Integration Marathon

Maybe something a bit easier:

 

HeroWise

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Re: HSC 2018 MX2 Integration Marathon

Damn that pi lemme sit on it a n=bit longer
 

fan96

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Re: HSC 2018 MX2 Integration Marathon

I put my solution as an attachment so that it won't spoil the answer for other people attempting to solve this.
 

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stupid_girl

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Re: HSC 2018 MX2 Integration Marathon

If you don't mind me asking, where do you get these integrals from?
Construct from simpler integrals using trig identities and properties of definite integrals.
 

stupid_girl

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Re: HSC 2018 MX2 Integration Marathon

Continue to have fun with trig.:jump:

Harder version:

Find the area between x-axis and y=f(x) on its maximal domain.

Simpler version:
Show that
 
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fan96

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Re: HSC 2018 MX2 Integration Marathon

Continue to have fun with trig.:jump:

Harder version:

Find the area between x-axis and y=f(x) on its maximal domain.

Simpler version:
Show that
a very nice integral.
 

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stupid_girl

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Re: HSC 2018 MX2 Integration Marathon

This one should be considerably easier than the previous one.:tongue:



The answer is pretty small. (1/32304)
 
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fan96

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Re: HSC 2018 MX2 Integration Marathon

I've reduced the integral to



if someone else wants to finish it from this, but it seems very difficult.

Maybe a different approach might be necessary?
 
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