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Looks like (A).

I think it would depend on the nature of the error.

Re: MATH1231/1241/1251 SOS Thread Substitute u = sinh(x), so du = cosh(x) dx. Then the integral becomes int (1+u^2) du = u + (u^3)/3 + c = sinh(x) + (sinh^3 (x))/3 + c.

Re: MATH1231/1241/1251 SOS Thread The main thing is that we need to split up the interval into cases based on where the thing in the absolute values is positive/negative in order to deal with the absolute values.

Re: MATH1231/1241/1251 SOS Thread $\noindent The integral has become $I = \int _{0}^{2\pi}\left|\cos \frac{x}{2} + \sin \frac{x}{2}\right|\, \mathrm{d} x = 2\int_{0}^{\pi} \left|\cos u + \sin u\right|\, \mathrm{d}u$ (substituting $u = \frac{x}{2}$). Let $f(u)=\cos u + \sin u$ for $u\in...

$\noindent Not sure exactly what type of Q. you mean. Something like find the coefficient of $x^{4}$ in the expansion of $\left(1+x^{2}\left)^3 \left(1+x\right)^5$? Finding the $x^0$ term is the easiest, it's just the product of the constant terms.$

You can generally just write "alternate segment theorem".

Re: MATH1231/1241/1251 SOS Thread $\noindent Need to use absolute value signs. Note that $1+\sin x = \cos^2 \frac{x}{2} + 2\sin \frac{x}{2} \cos \frac{x}{2} + \sin^2 \frac{x}{2}= \left(\cos \frac{x}{2} + \sin \frac{x}{2}\right)^2$. So $\sqrt{1+\sin x} = \left | \cos \frac{x}{2}+ \sin...

Re: HSC 2016 3U Marathon That's a classic method. Should be fine. (But probably safest to write it out with the binomial theorem formula (with the Sigma notation) before it, and/or say you're using the binomial theorem.)

$\noindent (It suffices to take the $\phi$ in my earlier post to be the \textsl{principal argument} $\mathrm{Arg}\left(a+bi\right)$. If $a$ is positive, then this is indeed just $\tan^{-1}\left(\frac{b}{a}\right)$. So if you wanted to always use $\tan^{-1}$, you could just factor out a negative...

You also generally need to be careful with the arctangent.

Re: First Year Uni Calculus Marathon Is there a typo in the first one? The last term in the product is 0, so that product is just 0 for all n.

Re: MATH1231/1241/1251 SOS Thread E is linear so E(Y) = -2E(X) + 3. For variance, use the property that Var(aX + b) = a2 Var(X), where a and b are constants. So here, Var(Y) = (-2)2Var(X) = 4Var(X).

Someone like that might say they of themself that they are "not a math person".

Re: MATH1231/1241/1251 SOS Thread $\noindent Remember there is a general probability rule that $\mathbb{P}\left(C\cup D\right) = \mathbb{P}\left(C\right) + \mathbb{P}\left(D\right) - \mathbb{P}\left(C\cap D\right)$ for events $C,D$. The solutions there typo'd, in the second thing you...

That's correct. :)

$\noindent It's the auxiliary angle transform. Just in general, we may write$ $$a \cos \theta + b\sin \theta = R\cos \left(\theta - \phi\right)$$, $\noindent where $R = \sqrt{a^{2} + b^{2}}$ and $\phi$ is the angle made by the point $(a,b)$ to the origin. In the setting of complex numbers (for...

Re: MATH1231/1241/1251 SOS Thread Yep to get 1/(1+x^3), replace x with x^3.

Re: MATH1231/1241/1251 SOS Thread Basically you should try and avoid having to derive the series unless you absolutely need to (or are specifically asked to). Just use well-known Taylor series to obtain the desired one if you can (like by replacing x with x^2 here). Regarding your...

Re: MATH1231/1241/1251 SOS Thread Just replace x with x^2. The series will be convergent for |x^2| < 1, i.e. -1 < x < 1.