Show that
Show that
Last edited by stupid_girl; 28 Oct 2018 at 10:28 PM.
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
Last edited by fan96; 11 Nov 2018 at 12:45 PM.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
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:
Last edited by Paradoxica; 11 Nov 2018 at 3:26 PM.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Maybe something a bit easier:
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
Damn that pi lemme sit on it a n=bit longer
Show that
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
I put my solution as an attachment so that it won't spoil the answer for other people attempting to solve this.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
Continue to have fun with trig.
Harder version:
Find the area between x-axis and y=f(x) on its maximal domain.
Simpler version:
Show that
Last edited by stupid_girl; 16 Dec 2018 at 10:32 PM.
This one should be considerably easier than the previous one.
The answer is pretty small. (1/32304)
Last edited by stupid_girl; 20 Dec 2018 at 7:31 PM.
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?
Last edited by fan96; 23 Dec 2018 at 2:01 PM.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
By using
and some manipulation, similar to the previous integral, we get
Because the integrand is even,
Integrating by parts,
Because for ,
This integral has been evaluated before to be equal to .
Last edited by fan96; 24 Dec 2018 at 3:16 PM.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
This one is absolutely a beast.
I have a question, When do you know to use
Last edited by HeroWise; 12 Jan 2019 at 5:41 PM.
This is another beast.
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
A tangent substitution will turn it into a format that Wolfram can solve...finally
https://www.wolframalpha.com/input/?...+(1-tan%5E2+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:
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.
Last edited by stupid_girl; 2 Feb 2019 at 8:23 PM.
This one may look simple at the first glance but actually trickier than you may have thought.
I'm sure a lot of people will come up with an answer 2.
Hint:
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
I think this is my first "Stupid_girl's Integrals" ill get. Maybe idk
Well didnt get a but got a Thats outta be good right?
I had to graph the thing, is it possible without graphing it?
Last edited by HeroWise; 3 Feb 2019 at 11:41 PM.
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