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September 15th, 2016, 07:55 PM   #1
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Substitution Required

I'm trying to find a non-trigonometric substitution with which to compute the integral
$$\int (x^2+1)^{-\frac32}\,\mathrm dx$$

Any thoughts?
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September 16th, 2016, 12:03 AM   #2
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put x=sh(t).
hyperbolic sinus.

use ch^2-sh^2=1.
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September 16th, 2016, 03:39 AM   #3
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The result is
$\frac{x}{\sqrt{x^2+1}}+C$

Last edited by abdallahhammam; September 16th, 2016 at 03:44 AM.
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September 16th, 2016, 07:56 AM   #4
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Quote:
Originally Posted by abdallahhammam View Post
The result is
$\frac{x}{\sqrt{x^2+1}}+C$
The derivative doesn't equal the integrand.
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September 16th, 2016, 09:40 AM   #5
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Hyperbolic substitution is also not allowed. I already knew the answer, it's the substitution that I needed.

I've found it now, so this becomes a challenge problem: solve the integral using a substitution that requires only the four basic operations and the square root.
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September 16th, 2016, 01:07 PM   #6
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my result is correct.

$(1+x^2)^{\frac{-3}{2}}=\frac{x^2+1-x^2}{(1+x^2)\sqrt{1+x^2}}$
the derivative of x/sqrt(1+x^2) is (1+x^2)^(-3/2).

you could also try the substitution x=1/t.

Last edited by abdallahhammam; September 16th, 2016 at 01:15 PM.
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September 16th, 2016, 01:45 PM   #7
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Does it work?
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September 16th, 2016, 01:52 PM   #8
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yes
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September 16th, 2016, 02:35 PM   #9
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I have to confess I didn't see how $x=\dfrac 1 t$ aided the evaluation of this integral.

Perhaps you can show your workings.
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September 16th, 2016, 02:43 PM   #10
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it becomes

$-\int (1+t^2)^{\frac{-3}{2}} t dt$
which is easier.
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