herrrrmmmmm (1 Viewer)

[Average Boreduser]

kk so yk how u can use hyperbolic trig subs to integrate stuff like 1/sqrtx^2+1 or smn? Are we allowed to just quote that arcsinh is ln(x+sqrt..wtv)? or do we hav to prove?

 

[liamkk112]

kk so yk how u can use hyperbolic trig subs to integrate stuff like 1/sqrtx^2+1 or smn? Are we allowed to just quote that arcsinh is ln(x+sqrt..wtv)? or do we hav to prove?

u wouldn't be asked those integrals at least in hsc, they're out of syllabus now i believe
and if they ever do, u can just write the ln(x+sqrt(x^2+1)) and show that differentiating that gives the integrand, no need for mentioning hyperbolic trig stuff

 

[Average Boreduser]

u wouldn't be asked those integrals at least in hsc, they're out of syllabus now i believe
and if they ever do, u can just write the ln(x+sqrt(x^2+1)) and show that differentiating that gives the integrand, no need for mentioning hyperbolic trig stuff

ah kk. Would I lose marks by using hyperbolic trig tho?

 

[WeiWeiMan]

kk so yk how u can use hyperbolic trig subs to integrate stuff like 1/sqrtx^2+1 or smn? Are we allowed to just quote that arcsinh is ln(x+sqrt..wtv)? or do we hav to prove?

why would you do that bruv

1. ∫ 1/sqrt(x^2+1) dx probably isn't showing up in the HSC (maybe in trials)
2. you can just multiply by some asspull on the top or do an asspull trig sub
x=tan(theta)
dx = sec^2(theta) dtheta
I = ∫ 1/sqrt(x^2+1)
= ∫sec^2(theta)/sec(theta) dtheta
= ∫ sec(theta)dtheta
= ∫ sec^2(theta)+sec(theta)tan(theta) / sec(theta)+tan(theta) d/theta
= ln|sectheta+tan(theta)|+C
= ln|x + sqrt(x^2+1)|+C

prob just mutliply by smth on the top or memorise ngl

 

[Average Boreduser]

why would you do that bruv

1. ∫ 1/sqrt(x^2+1) dx probably isn't showing up in the HSC (maybe in trials)
2. you can just multiply by some asspull on the top or do an asspull trig sub
x=tan(theta)
dx = sec^2(theta) dtheta
I = ∫ 1/sqrt(x^2+1)
= ∫sec^2(theta)/sec(theta) dtheta
= ∫ sec(theta)dtheta
= ∫ sec^2(theta)+sec(theta)tan(theta) / sec(theta)+tan(theta) d/theta
= ln|sectheta+tan(theta)|+C
= ln|x + sqrt(x^2+1)|+C

prob just mutliply by smth on the top or memorise ngl

I mean its quicker tho if u were allowed to just quote it (arcsinh). Bc its kinda a function so I was thinking like- surely we can quote it

 

[Average Boreduser]

bc arcsinh^2-arccosh^2=1 and manipulating etcetc makes it like a 2 line proof

 

[lqmoney]

I wouldn’t go out of syllabus unless you can’t get a question using standard techniques. There are lots of questions in the HSC you do faster with out of syllabus stuff but you would lose marks for not showing working. For these integrals a fast way is rewriting the integrand as ((x+sqrt(x^2+a))/sqrt(x^2+a))/(x+sqrt(x^2+a) which can be simplified as ((x/sqrt(x^2+a))+1)/(sqrt(x^2+a)+x) which is of the form f’(x)/f(x)

 

[liamkk112]

ah kk. Would I lose marks by using hyperbolic trig tho?

prob not but it's out of syllabus so to be safe either just quote the log result if you're bothered to remember it and prove it by differentiating, or just use substitution
i wouldn't really advise writing any of the hyperbolic trig stuff