Integration - FP2 Watch

Wish
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#1
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\int \frac{7x}{\sqrt{16-x^4}} \ \mathrm{d}x

Using substitution u = x^2 to obtain \frac 72 \int \frac 1{\sqrt{16 - u^2}} \ \mathrm{d}u.

Arcsin Rule: \int \frac{1}{\sqrt{a^2 - x^2}} \ \mathrm{d}x = \mathrm{arcsin}\left(\frac{x}{a}  \right)

Gives me answer \frac72 \mathrm{arcsin}\left(\frac{x^2}{  4}\right)

But according to integrator, it is \csc ^{-1}. http://integrals.wolfram.com/index.j...9&random=false

Whats wrong?
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insparato
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What does it say in the book ? Shouldn't it give you the answer  \frac{7}{2} arcsin \frac{x^2}{4} ? Apart from that it looks fine to me.
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Wish
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(Original post by insparato)
What does it say in the book ? Shouldn't it give you the answer  \frac{7}{2} arcsin \frac{x^2}{4} ? Apart from that it looks fine to me.

Sorry, typo, I meant  \frac{7}{2} arcsin \frac{x^2}{4} . Integrator is giving a different answer.
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generalebriety
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(Original post by Wish)
\int \frac{7x}{\sqrt{16-x^4}} \ \mathrm{d}x

Using substitution u = x^2 to obtain \frac 72 \int \frac 1{\sqrt{16 - u^2}} \ \mathrm{d}u.

Arcsin Rule: \int \frac{1}{\sqrt{a^2 - x^2}} \ \mathrm{d}x = \mathrm{arcsin}\left(\frac{x}{a}  \right)

Gives me answer \frac72 \mathrm{arcsin}\left(\frac{4}{x^  2}\right)

But according to integrator, it is \csc ^{-1}. http://integrals.wolfram.com/index.j...9&random=false

Whats wrong?
Don't you mean 7/2 arcsin(x^2 / 4)? :confused:

It can also be written in the way integrator wrote it. It's the same thing - check by differentiating both if you don't believe it. After all, if sin y = a then cosec y = 1/a, so arcsin(a) = arccosec(1/a). Put a = x^2 / 4...

Edit: typo.
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DavyS
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cosec(x) = sin(1/x)
Edit: Duh..

\mathrm{arccosec}x = \mathrm{arcsin}(\frac{1}{x})

Therefore, your answer, after insparato's correction, and the integrators are equivalent.
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Wish
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#6
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Got it, Thanks!
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insparato
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#7
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Aha, Interesting. Never come across that before.
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Zii
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(Original post by DavyS)
cosec(x) = sin(1/x)

Therefore, your answer, after insparato's correction, and the integrators are equivalent.
Erm, no.

\mathrm{cosec}x \not= \mathrm{sin}(\frac{1}{x}), \mathrm{cosec}x = \frac{1}{\mathrm{sin}x}.
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DavyS
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(Original post by Zii)
Erm, no.

\mathrm{cosec}x \not= \mathrm{sin}(\frac{1}{x}), \mathrm{cosec}x = \frac{1}{\mathrm{sin}x}.
-hits self on head very hard-

You should have been able to pick up my error from the context, no?

\mathrm{arccosec}x = \mathrm{arcsin}(\frac{1}{x})
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generalebriety
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(Original post by DavyS)
-hits self on head very hard-

You should have been able to pick up my error from the context, no?

\mathrm{arccosec}x = \mathrm{arcsin}(\frac{1}{x})
Problem is that helpers often get irritated when other helpers correct them, because it was only a silly mistake. You knew what you were talking about as much as the person who corrected you, but occasionally that can confuse the people who were asking for help in the first place. Remember who you're helping here.
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DavyS
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(Original post by generalebriety)
Problem is that helpers often get irritated when other helpers correct them, because it was only a silly mistake. You knew what you were talking about as much as the person who corrected you, but occasionally that can confuse the people who were asking for help in the first place. Remember who you're helping here.
Oh, I didn't mean to come across as annoyed at all. I totally understand the confusion I could have caused and stuff, and that was honestly just a stupid mistake which I corrected as soon as it came to my attention.

That was less an outcry at all those with the audacity to correct me, than an admission of stupidity and not checking my post well enough :P
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Zii
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(Original post by DavyS)
Oh, I didn't mean to come across as annoyed at all. I totally understand the confusion I could have caused and stuff, and that was honestly just a stupid mistake which I corrected as soon as it came to my attention.

That was less an outcry at all those with the audacity to correct me, than an admission of stupidity and not checking my post well enough :P
Yeah don't worry about it, I just like correcting people. Hell, I get corrected often enough!

Silly mistakes are the worst :p: .
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