anonymous
  • anonymous
Evaluate the Intergal
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
\[\int\limits\limits_{}^{}\frac{ \arcsin(x) }{ x^2 } dx\]
anonymous
  • anonymous
I have no idea... I am think about trig substitutions but I am not sure about this...
anonymous
  • anonymous
Not sure either, but what does \(x=\sin(u)\) get you?

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anonymous
  • anonymous
Why would you do that?
AccessDenied
  • AccessDenied
I was thinking about trying integration by parts... have you tried that yet? :)
anonymous
  • anonymous
It would get rid of the arcsin.
anonymous
  • anonymous
It's just a guess
anonymous
  • anonymous
Well I was going to make a substitution and then use intergration by parts but I am not sure...
anonymous
  • anonymous
What if I said u=arcsin(x) ?
anonymous
  • anonymous
That's the same as x = sin(u)
anonymous
  • anonymous
Right. But if I said that I would get a squart root and I could use trigonometric substitution.
anonymous
  • anonymous
Allright so I so far got... Use intragtion by parts: u=arcsinx) \[du=\frac{ 1 }{ \sqrt{1-x^2} }\] dv=x^-2 dx v=-x^-1
anonymous
  • anonymous
\[\frac{ -\arcsin(x) }{ x }-\int\limits_{}^{}\frac{ 1 }{ x \sqrt{1-x^2} }dx\]
anonymous
  • anonymous
I am kinda stuck at this point.
BAdhi
  • BAdhi
First do the substitution $$x=\sin(u)\implies dx=\cos(u)du$$ then $$\int \frac{\arcsin{x}}{x^2}\;dx=\int \frac{u\cos(u)}{\sin^2(u)}\;du=\int u\csc(u)\cot(u)\;du$$ $$\int u\frac{d(-\csc(u))}{du}du$$ do the integration by parts
anonymous
  • anonymous
Hmm let me see...
anonymous
  • anonymous
@BAdhi So I got to this point \[\int\limits_{}^{}ucsc(u)\cot(u) du\]
anonymous
  • anonymous
I would use integration by parts here right?
BAdhi
  • BAdhi
we know that $$\frac{d\csc(u)}{du}=-\csc(u)\cot(u)$$ then, $$\begin{align*}\int u\csc(u)\cot(u)\,du&=\int u\frac{-d\csc(u)}{du}\,du\\ &=-u\csc(u)+\int \csc(u)\,du \end{align*}$$
anonymous
  • anonymous
Yep I got that far.
anonymous
  • anonymous
Now I need to figure out the integral of csc(x).
anonymous
  • anonymous
Thanks I got this :) .
anonymous
  • anonymous
could use x = sin(u)
anonymous
  • anonymous
Thank you too wio :) .

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