@satellite73

- angela210793

@satellite73

- jamiebookeater

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- angela210793

|dw:1341334653707:dw|

- anonymous

ick
i think the idea is to write the denominator as a single trig function

- angela210793

:O i thought maybe to split and take 2 fractions O.o

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## More answers

- angela210793

but idk how to do tht -_-

- 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

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

i don't think there is a snappier way to do this. let me think for a second

- angela210793

okok

- 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

hmmmm....suppose i know tht....how to use it O.o

- 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

don't worry about the arctan part, that is a number

- anonymous

so all you need to do is look up the anti derivative of cosecant and you are done

- anonymous

here is a video explanation
http://www.youtube.com/watch?v=STUh3ni4l50

- angela210793

hmmmm....we've never use secant...wht is it?

- anonymous

cosecant
it is the reciprocal of sine

- 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

:/....ok..
thank you Sir :D

- anonymous

maybe we can get some help
i will repost there might be a snappy trick

- angela210793

okk

- anonymous

maybe myininaya has a snappy way
multiply top and bottom by \(3\sin(x)+4\cos(x)\) maybe?

- myininaya

I don't know @satellite73 I like what you did.

- anonymous

ok then i will stick to that. thanks

- myininaya

I actually think what sat did was probably the most snappiest thing you can do for this type of integral

- anonymous

If we replace sinx by 2tan(x/2)/(1 + tan^2(x/2),

- angela210793

sec=1/cosx????

- myininaya

yes @angela210793
sec(x)=1/(cos(x))
csc(x)=1/(sin(x))

- angela210793

ok....thanks guys :D

- 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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