September 15th, 2016, 07:55 PM  #1 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra  Substitution Required
I'm trying to find a nontrigonometric substitution with which to compute the integral $$\int (x^2+1)^{\frac32}\,\mathrm dx$$ Any thoughts? 
September 16th, 2016, 12:03 AM  #2 
Senior Member Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 
put x=sh(t). hyperbolic sinus. use ch^2sh^2=1. 
September 16th, 2016, 03:39 AM  #3 
Senior Member Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 
The result is $\frac{x}{\sqrt{x^2+1}}+C$ Last edited by abdallahhammam; September 16th, 2016 at 03:44 AM. 
September 16th, 2016, 07:56 AM  #4 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  
September 16th, 2016, 09:40 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
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. 
September 16th, 2016, 01:07 PM  #6 
Senior Member Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 
my result is correct. $(1+x^2)^{\frac{3}{2}}=\frac{x^2+1x^2}{(1+x^2)\sqrt{1+x^2}}$ 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. 
September 16th, 2016, 01:45 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
Does it work?

September 16th, 2016, 01:52 PM  #8 
Senior Member Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 
yes

September 16th, 2016, 02:35 PM  #9 
Senior Member Joined: Sep 2015 From: USA Posts: 2,501 Thanks: 1372 
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. 
September 16th, 2016, 02:43 PM  #10 
Senior Member Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 
it becomes $\int (1+t^2)^{\frac{3}{2}} t dt$ which is easier. 

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