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.
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