Difficult Integral

  • Thread starter iRaid
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  • #26
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Well, you could also have started with a straight forward substitution: ##y=\arcsin x##
This means that ##x=\sin y##.
You're rid of that pesky arcsin and get easier functions to integrate.

You'd have:
$$\int (\arcsin x)^2 dx = \int y^2 d(\sin y)$$

Now do integration by parts:
$$\int y^2 d(\sin y) = y^2 \sin y - \int 2y \cos y dy$$

Repeat integration by parts, and after that back substitute x for y...

Sorry I just looked back at this thread, this is interesting, but I don't know how to follow it. Could someone explain?
 
  • #27
Dick
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Sorry I just looked back at this thread, this is interesting, but I don't know how to follow it. Could someone explain?

What part don't you get? Do you have a problem with ILS's clever substitution?
 
  • #28
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What part don't you get? Do you have a problem with ILS's clever substitution?

I'm not sure on his notation with the d(siny) part particularly.
 
  • #29
Dick
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I'm not sure on his notation with the d(siny) part particularly.

ILS just substituted x=sin(y). So dx becomes d(sin(y)). arcsin(x)=arcsin(sin(y))=y. dx=d(sin(y))=cos(y)dy. You might have noticed in the way I led you through there was a lot of repetition in calculating the various integrals. ILS's idea compacts that a bit.
 
  • #30
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ILS just substituted x=sin(y). So dx becomes d(sin(y)). arcsin(x)=arcsin(sin(y))=y. dx=d(sin(y))=cos(y)dy. You might have noticed in the way I led you through there was a lot of repetition in calculating the various integrals. ILS's idea compacts that a bit.

I see it, thanks. But I would probably never think of that lol.
 
  • #31
Dick
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I see it, thanks. But I would probably never think of that lol.

Well, I didn't either. Don't feel bad.
 
  • #32
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Well, I didn't either. Don't feel bad.

:) Thanks for the help, I appreciate it.
 

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