angela210793
  • angela210793
@satellite73
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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angela210793
  • angela210793
|dw:1341334653707:dw|
anonymous
  • anonymous
ick i think the idea is to write the denominator as a single trig function
angela210793
  • angela210793
:O i thought maybe to split and take 2 fractions O.o

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angela210793
  • angela210793
but idk how to do tht -_-
anonymous
  • anonymous
it is going to be \(5\sin(x+\theta)\) but i don't see a nice form for \(\theta=\tan^{-1}(\frac{4}{3})\)
anonymous
  • anonymous
actually it doesn't matter because \(\theta\) is a constant also i made a mistake, it is \(5\sin(x-\theta)\) still no matter
anonymous
  • anonymous
i don't think there is a snappier way to do this. let me think for a second
angela210793
  • angela210793
okok
anonymous
  • anonymous
ok no i can't think of a better way, and also i tried wolfram and what a disaster what you need to know is \[a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\theta)\] where \(tan(\theta)=\frac{b}{a}\)
angela210793
  • angela210793
hmmmm....suppose i know tht....how to use it O.o
anonymous
  • anonymous
well then it is easy, especially if you have a table of integrals this becomes \[\frac{1}{5}\int\csc\left(x-\tan^{-1}(\frac{4}{3})\right)\]
anonymous
  • anonymous
don't worry about the arctan part, that is a number
anonymous
  • anonymous
so all you need to do is look up the anti derivative of cosecant and you are done
anonymous
  • anonymous
here is a video explanation http://www.youtube.com/watch?v=STUh3ni4l50
angela210793
  • angela210793
hmmmm....we've never use secant...wht is it?
anonymous
  • anonymous
cosecant it is the reciprocal of sine
anonymous
  • anonymous
integral is \[\int csc(x)dx=-\ln(\cot(x)+\csc(x))\] if you have not used this i really have no idea how you are supposed to do this problem maybe multiply top and bottom by the conjugate?
angela210793
  • angela210793
:/....ok.. thank you Sir :D
anonymous
  • anonymous
maybe we can get some help i will repost there might be a snappy trick
angela210793
  • angela210793
okk
anonymous
  • anonymous
maybe myininaya has a snappy way multiply top and bottom by \(3\sin(x)+4\cos(x)\) maybe?
myininaya
  • myininaya
I don't know @satellite73 I like what you did.
anonymous
  • anonymous
ok then i will stick to that. thanks
myininaya
  • myininaya
I actually think what sat did was probably the most snappiest thing you can do for this type of integral
anonymous
  • anonymous
If we replace sinx by 2tan(x/2)/(1 + tan^2(x/2),
angela210793
  • angela210793
sec=1/cosx????
myininaya
  • myininaya
yes @angela210793 sec(x)=1/(cos(x)) csc(x)=1/(sin(x))
angela210793
  • angela210793
ok....thanks guys :D
anonymous
  • anonymous
\[\huge sinx = \frac{2\tan \frac{x}{2}}{1+ \tan^2\frac{x}{2}}\] \[\huge cosx = \frac{1 - \tan^2\frac{x}{2}}{1+ \tan^2\frac{x}{2}}\]

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