Calculus problem. Help for a medal! :)
(posting below)

Question

Calculus problem. Help for a medal! :)
(posting below)

anonymous · Answer

$$\int\limits_{}^{}(\csc(2x))dx$$

anonymous · Answer

I decided that...
u= 2x
du=2dx

samigupta8 · Answer

Do u know the integral of cosec x ?

anonymous · Answer

I think it's -csc(x)cot(x).

anonymous · Answer

or maybe that's the derivative?

samigupta8 · Answer

Dis is not its integral ...u r telling me the differential form of ir...

samigupta8 · Answer

Exactly the latter one...

samigupta8 · Answer

Anyways the integral is 
ln(cosecx-cotx)

anonymous · Answer

How do you get to that answer?

samigupta8 · Answer

Ok...i will show u the work...

samigupta8 · Answer

It's like u can multiply n divide  
cosecx dx by (cosecx-cotx)

samigupta8 · Answer

Cosecx(cscx-cotx)dx/cscx-cotx

samigupta8 · Answer

Like dis...

anonymous · Answer

Is that a trig rule?

samigupta8 · Answer

For simplicity purpose i have used this rule...

samigupta8 · Answer

Now u can open up the bracket in numerator term n the equation u will get is csc^2x-cscx cotx

samigupta8 · Answer

Did u get it till here?

anonymous · Answer

just a moment

anonymous · Answer

If I were to show this is the form of u-substitution, what exactly would I make u?

samigupta8 · Answer

I m telling u the integral of cosec x ....if u know that u can solve dis problem easily

anonymous · Answer

Okay.

samigupta8 · Answer

So u got it till now

anonymous · Answer

I believe so

samigupta8 · Answer

Gud...

samigupta8 · Answer

Now what u have to do is jst put the denominAtor term which is cscx-cotx as t ....then differentiate it. Like
 cscx-cotx=t
(-Cscxcotx+csc^2x)dx=dt...

samigupta8 · Answer

Did u get it?

anonymous · Answer

Somewhat. Just keep going, and it'll probably make more sense.

samigupta8 · Answer

Oh...that's absolutely fine vid me...

samigupta8 · Answer

Okk so now this term dt is exactly matching with our numerator term of integration.... So now jst replace that by dt n the denominator by t....

samigupta8 · Answer

So finally u will b left vid integration of dt/t which is lnt....

samigupta8 · Answer

Now put the value of t back in this equation which is cscx-cotx...

samigupta8 · Answer

So ans will be ln(cscx-cotx)

anonymous · Answer

So it'd be... $$\frac{1}{2}\ln (\csc(2x)-\cot(2x))$$

samigupta8 · Answer

Exactly u got it ryt....

anonymous · Answer

Oops I forgot the "-" on the 1/2   and the "+ C"

samigupta8 · Answer

Y a -ve sign?

anonymous · Answer

Well I have an answer document, that says there's a "-" sign. I'm not 100% sure why though.

samigupta8 · Answer

Actually the integral of cscx-cotx is ln|cscx-cotx|

anonymous · Answer

So there's no reason there should be a "-"?

anonymous · Answer

Okay I figured out why there's a negative sign... https://www.youtube.com/watch?v=byB0Vz3dcsE

welshfella · Answer

wolfram gives the following result http://www.wolframalpha.com/input/?i=Integrate+++cosec+%282x%29

anonymous · Answer

Symbolab also gives... https://www.symbolab.com/solver/integral-calculator/%5Cint%5Cleft(csc%5Cleft(2x%5Cright)%5Cright)dx/?origin=enterkey So I have no idea what's right.

welshfella · Answer

sometimes you can get different integrals but both are right

welshfella · Answer

wolfram used the substitution u = 2x

anonymous · Answer

My book gives me 
$$-\frac{1}{2}\ln|\csc(2x)+\cot(2x)|+C$$ Do you know how to get this answer?

anonymous · Answer

Specifically

welshfella · Answer

yea i got that in my book too!

anonymous · Answer

What book do you have?

welshfella · Answer

i think the wolfram method is easier

yoy need to integrate 
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