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#1 2019-05-12 11:45:12

Anthony Lahmann
Member
Registered: 2019-04-25
Posts: 6

Integral of 1/sqrt(1-x^2)

I will integrate 1/sqrt(1-x^2) by u-substitution. Here's how I did it:


After we did the u-substitution, we end up with the exact same integral, but with a negative in the front. What happened?

EDIT 1: As bob bundy pointed out, there may be something wrong with this. I double checked, and it turns out nothing is wrong.

Last edited by Anthony Lahmann (2019-05-13 12:36:59)

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#2 2019-05-12 19:01:02

bob bundy
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Registered: 2010-06-20
Posts: 8,594

Re: Integral of 1/sqrt(1-x^2)

hi Anthony Lahmann

Welcome to the forum. 

According to Wolfram Alpha, this is sin-1(x) and the second is -sin-1(x).  So there's a sign error in there somewhere.  I'll try to track it down later.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#3 2019-05-12 22:37:24

zetafunc
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Registered: 2014-05-21
Posts: 2,249
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Re: Integral of 1/sqrt(1-x^2)

Anthony Lahmann wrote:

I will integrate 1/sqrt(1-x^2) by u-substitution. Here's how I did it:


After we did the u-substitution, we end up with the exact same integral, but with a negative in the front. What happened?

Your issue is here:

The correct implication is:

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#4 2019-05-13 12:23:15

Anthony Lahmann
Member
Registered: 2019-04-25
Posts: 6

Re: Integral of 1/sqrt(1-x^2)

bob bundy wrote:

hi Anthony Lahmann

Welcome to the forum. 

According to Wolfram Alpha, this is sin-1(x) and the second is -sin-1(x).  So there's a sign error in there somewhere.  I'll try to track it down later.

Bob

There is nothing wrong with this! I differentiated the u correctly, and I solved for x correctly.

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#5 2019-05-14 00:00:27

George,Y
Member
Registered: 2006-03-12
Posts: 1,345

Re: Integral of 1/sqrt(1-x^2)

Anthony Lahmann wrote:

There is nothing wrong with this! I differentiated the u correctly, and I solved for x correctly.

the last change of u to x is your mistake.


X'(y-Xβ)=0

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#6 2019-05-14 00:35:11

Alg Num Theory
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Registered: 2017-11-24
Posts: 693
Website

Re: Integral of 1/sqrt(1-x^2)

bob bundy wrote:

According to Wolfram Alpha, this is sin-1(x) and the second is -sin-1(x).

Funny – I wonder why it didn’t give the second one as cos⁻¹(x).

If you integrate correctly, you should get:

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[*]

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If you think the results are strange, remember the identity:

[list=*]
[*]

.[/*]
[/list]

Also when you integrate, do not forget to include an arbitrary constant.


Me, or the ugly man, whatever (3,3,6)

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