Relax I wanted to use my partial fractions method.lel noob
 \sqrt{x^2 + 2}} \, $d$x = \int_{\sqrt{2}}^{\sqrt{3}} \frac{v\text{d}v}{(v^2+1)|v|} = \int_{\sqrt{2}}^{\sqrt{3}} \frac{\text{d}v}{v^2+1} \because \, v>0 \\ \int_0^1 \frac{x}{(x^2 + 3) \sqrt{x^2 + 2}} \, $d$x = \frac{\pi}{3}-\tan^{-1}\sqrt{2})
For some reason, I seem to lose a factor of 2 during the final substitution (meaning that my final answer is = \frac{1}{x^4+1} \\ $a) By considering the substitution $y = \frac{1}{x}$ show that $ \int_{-\infty}^{\infty} f(x)$d$x = \int_{-\infty}^{\infty} x^2 f(x)$d$x. \\ $b) Using the substitution $t = x-\frac{1}{x}$, evaluate the integral.$)
Yeah, and the answer that I get for 2I (, not $I$.$)
Yeah, and the answer that I get for 2I () (apparently) isn't right; wolfram alpha confirms that I should get
as my answer for 2I
Ah, thankyou, I should have seen to do that. (especially given that I did a similar thing in part a)
I could not see a feasible substitution, but the indexes were suggestive of a quotient of fifth roots, so that's what I went for.
^{\frac{6}{5}} (x - 7)^{\frac{4}{5}}} = \frac{5}{9}\left(\frac{x-7}{x+2}\right)^{\frac{1}{5}}+\mathcal{C}$)
It is perhaps a bit much to expect a current student to tackle the above integral without hints, so here is a a rough walkthrough:
You definitely would have, but anyway you don't need to have heard of or seen them before to do this question.
Very possible with HSC knowledge, especially with the number of given hints.
Nobody actually knows what you mean when you refer to "inequalities from repeated integration". You need to provide an explanation for this vague and ambiguous claim.
Maybe referring to the Taylor polynomial bounds? E.g. things like
Too bad that's outside MX2 scopeMaybe referring to the Taylor polynomial bounds? E.g. things like
.
See, I already knew this conceptually in my head, but no labels are ever placed on my ideas unless they are already labelled in the external world.Apologies for vagueness, didn't want to hand-hold too much as I think it is a nice thing for a student to discover by him/herself. I will elaborate now
Yep, Integrand is exactly right. I was referring to the upper and lower bounds coming from the alternating partial Taylor sums.
That knowing anything about Taylor series is out of syllabus is irrelevant, as you don't need to have seen these before and you don't need to know what these things are called.
It is MX2 knowledge thatfor non-negative real x.
This implies thatfor non-negative real x.
This implies thatfor non-negative real x.
etc.
The inequalities arising from iterating this process are exactly those I was referring to with my statement about repeated integration.
Note that combining the above inequalities with what we already know from syllabus, we have:
for non-negative x, and
We can obtain "tighter" upper and lower bounds for sin and cos by iterating this process. I leave it to you guys to figure out exactly how far you need to go.
Also, as a slight aside: I am sure most of you realise from syllabus knowledge that quantities like sin(cx)/x and sin(cx)/sin(x) tend to c as x tends to zero. For this reason, when we write something like sin(cx)/x in this question, you might as well assume that we are instead referring to the function f(x) that is equal to sin(cx)/x for nonzero x and has f(c)=c. f is continuous at 0 so it is more convenient notationally to do this.