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HSC 2018-2019 MX2 Integration Marathon (1 Viewer)

stupid_girl

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For this integral, some trig identities may make your life easier. Of course, Weierstrass substitution will also work.




 

stupid_girl

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#83 and #88 are still outstanding and this is a new one.
Feel free to share your attempt.
It seems no one has attempted yet. This one is a little bit interesting.
At the first glance, everything is related to 4^x so it makes sense to substitute u=4^x to get


If you put these two functions for 1<u<4 in a graphing software, you should be able to sense what's going on.



The area should add up to a 3x3 square, ie.
 
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Drdusk

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She needs to change her name. The irony is so real...
 

HeroWise

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Funny thing is i only got out like 3 of them out and sacked the rest.

Yeah we are the stupid ones ahha
 

stupid_girl

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This is a rather tough one...especially the final simplification. Feel free to share your attempt.
 

fan96

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For , let



Prove that



Given that



Show that



 

Daniel.22

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This is a rather tough one...especially the final simplification. Feel free to share your attempt.
It is just tedious...

Edit: Sorry, typoed the numerator of the second integral in the second line of the evaluation of I, should just be cos(2x).

26866
 
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Daniel.22

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For , let



Prove that



Given that



Show that



I'll leave the intended solution for someone else to write out (p.s I think you need to be a bit careful about the first part, the integrals are indefinite so equality will only hold up to a constant as written.)

A cute alternate solution (but I don't believe this technique is allowed in MX2):

26867
 

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