anonymous
  • anonymous
Help with integrals
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
what do you need help with?
anonymous
  • anonymous
\[\int\limits_{}^{}\frac{dx}{\sin^2x}\]
anonymous
  • anonymous
I got stuck here :\[2\int\limits_{}^{}\frac{dx}{1-\cos(2x)}\]

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anonymous
  • anonymous
hold on let me write it out ok?
anonymous
  • anonymous
but btw the 1-cos(2x) use trig identitites.
anonymous
  • anonymous
sure. thanks
anonymous
  • anonymous
So how is it?
anonymous
  • anonymous
\[\int\limits 1\div \sin x ^{2} dx = x / sinx ^{2}\]
anonymous
  • anonymous
Nah.
anonymous
  • anonymous
-1/tan x sorry wrong answer.
anonymous
  • anonymous
How did you get it?
anonymous
  • anonymous
It's normally assumed from the fact \[\frac{\mathbb{d}}{\mathbb{d}x} \cot x = -cosec^2x\] .... but if you REALLY want to prove it, try the sub \[t = \tan \frac{x}{2} \] Haven't tried it, but it almost always works. May be some easy way I'm missing.
anonymous
  • anonymous
Note, proving the differentiation version, if you deem that sufficient, is far easier. But if you did not know the result, the sub I guess would be OK.
anonymous
  • anonymous
moolean, i'd stick with what hes saying, i think he understand integrals better then i do. im nto perfect at it yet, i was just trying to help, but he seems to rlly know what he's talking about.
anonymous
  • anonymous
\[\text{Let } t = \frac{\tan{x}}{2} \implies \frac{\mathbb{d}x}{\mathbb{d}t} = \frac{2}{1+t^2} \] It quickly follows that: \[\sin x = \frac{2t}{1+t^2}\] \[\int \frac{1}{sin^2x} \mathbb{d}x = \int \left(\frac{1+t^2}{2t}\right)^2\cdot \frac{2}{1+t^2} \mathbb{d}t = \int \left(\frac{1+t^2}{2t^2}\right)\cdot \mathbb{d}t \] \[\int \left(\frac{1+t^2}{2t^2}\right)^2\cdot \mathbb{d}t = \frac{t^2 - 1}{2t} \] Which, from double angle formulae, is equal to the required result.
anonymous
  • anonymous
It did work, after all :D
anonymous
  • anonymous
gj Newton :P that's impressive.
anonymous
  • anonymous
Thanks - I needed to practice it anyway. NOTE there should be no squared on the fraction on the last line. some dodgy copy/pasting from above there :(
anonymous
  • anonymous
THAT;S A GREAT HELP. THANKS!
anonymous
  • anonymous
Ugh, the sub is \[ t = \tan \frac{x}{2} \] So many typos :/ But you're welcome.
anonymous
  • anonymous
geez i should say thank you too, u taught me something and i was only trying to help :P

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