Recent content by tywebb

Project Academy have a pdf here : https://drive.google.com/file/d/1g7PhX9K1YLB972zFu83kqdM1hVb6RklP/view It came out in 1967 in the Coroneos and Lynch Level 1 text and then in 1981 in the dumbed down 4 unit text when 4 unit changed from a 2-year course to a 1-year course. I think however that...

This is from the old Coroneos 100 Question 17 There are somewhat inefficient methods of solution, such as in Lumi's 2008 solutions. This is somewhat unsatisfactory. In the context of the old syllabus there was a standard integral sheet, some of which are no longer included in the current...

|z|=\sqrt{\cos^4\theta+\sin^4\theta}=\sqrt{1-\frac{1}{2}\sin^22\theta} 0\le\sin^22\theta\le1\therefore\frac{1}{2}\le1-\frac{1}{2}\sin^22\theta\le1 \therefore\frac{1}{\sqrt2}\le|z|\le1 B

alternatively, u know \tan\theta-\sec\theta=1-\frac{2}{\tan\frac{\theta}{2}+1} and so \int_0^\frac{\pi}{2}\frac{1}{\sin\theta+1}\ \!d\theta=\int_0^\frac{\pi}{2}(\sec^2\theta-\sec\theta\tan\theta)\...

yeah u can actually did u know that \tan\theta-\sec\theta=\sin\left(\frac{\theta}{2}-\frac{\pi}{4}\right)\csc\left(\frac{\theta}{2}+\frac{\pi}{4}\right) whereupon we may do this one without the limits \int_0^\frac{\pi}{2}\frac{1}{\sin\theta+1}\...

now we redo the original question. i don't think at this stage we need to give up on your method. it works! but just other way around. \int_0^\frac{\pi}{2}\frac{1}{\sin\theta+1}\ \!d\theta=\int_0^\frac{\pi}{2}(\sec^2\theta-\sec\theta\tan\theta)\...

u know why this? i show u now \begin{aligned}\lim_{\theta\rightarrow\frac{\pi}{2}}(\tan\theta-\sec\theta) &=\lim_{\theta\rightarrow\frac{\pi}{2}}\frac{\sin\theta-1}{\cos\theta}\\ &=\lim_{\theta\rightarrow\frac{\pi}{2}}\frac{\frac{d}{d\theta}(\sin\theta-1)}{\frac{d}{d\theta}\cos\theta}\\...

technically in the context of this integral one only needs to do the left-sided limit \lim_{\theta\rightarrow\(\frac{\pi}{2})^-}(\tan\theta-\sec\theta)=0 but it works from both sides anyway look at graph of y=tan x-sec x u can see it approaches 0 as x->pi/2

u can do limit \lim_{\theta\rightarrow\frac{\pi}{2}}(\tan\theta-\sec\theta)=0

here another way \int_0^\frac{\pi}{2}\frac{1}{\sin\theta+1}\ \!d\theta=\int_0^\frac{\pi}{2}\frac{1}{2}\csc^2\left(\frac{\theta}{2}+\frac{\pi}{4}\right)\ \!d\theta=1

It's the other way around \frac{1}{\sin\theta+1}=\sec^2\theta-\tan\theta\sec\theta Answer is 1 and \int_0^\frac{\pi}{2}(\sec^2\theta-\tan\theta\sec\theta)\ \!d\theta=1 so method is ok, but just do it the other way around

as i said before it is out-of-print it didn't come with ebook either nevertheless a small part of it is available online - lumi's solutions to the famous "coroneos 100": https://4unitmaths.com/coroneos100-lumi-sol.pdf

That's great! Are you aware that this has been done before? - by Jose Lumi back in 2008 but I think it's out-of-print now i have them though