What is the most efficient integration technique for (1 + x^2)^(1/2) dx?

  • Context: Graduate 
  • Thread starter Thread starter arevolutionist
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary

Discussion Overview

The discussion centers around identifying the most efficient integration technique for the integral of (1 + x^2)^(1/2) dx. Participants explore various methods, including trigonometric and hyperbolic substitutions, and share their reasoning regarding the effectiveness of these approaches.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using trigonometric substitution for the integral.
  • Another participant counters that trigonometric substitutions are not suitable, proposing instead that hyperbolic substitution would be more effective, specifically using the relationship cosh^2(y) - sinh^2(y) = 1.
  • A different approach is mentioned involving the substitution x = tan(m), which leads to an integral of sec^3(m) dm, suggesting that integration by parts could be applied.
  • Concerns are raised about the complexity of integrating sec^3, with one participant noting that a hyperbolic substitution may be quicker.

Areas of Agreement / Disagreement

Participants express differing opinions on the best integration technique, with no consensus reached on which method is the most efficient.

Contextual Notes

Participants reference various integration techniques without resolving the effectiveness of each method, indicating that the discussion is exploratory and contingent on individual preferences and experiences.

arevolutionist
Messages
60
Reaction score
0
Which integration technique should I use for something similar to:

(1 + x^2)^(1/2) dx
 
Physics news on Phys.org
Trig substitution, I'd say.
 
No. All of the "trig substutions" are based on sin^2+ cos^2= 1 and so involve either "1-" or "-1".

Oh, now that was a silly thing for me to say! sin^2(x)+ cos^2(x)= 1 so, dividing through by cos^2(x), we have
tan^2(x)+ 1= sec^2(x), just as Sleek suggested. (I am delighted that, while pointing out that I was completely wrong, he referred to an earlier post of mine!)

Looks to me like a hyperbolic substition should work. Since cosh^2(y)- sinh^2(y)= 1, cosh^2(y)= 1+ sinh^2(y). Let x= sinh(y).
 
Last edited by a moderator:
Even x=tan(m) would work. One would end up with int([sec(m)]^3) dm. This can be integrated using Int By Parts by differentiating sec(x) and integrating sec^2(x). Also, it can be integrated by method HallsofIvy suggested here: https://www.physicsforums.com/archive/index.php/t-156162.html.

Regards,
Sleek.
 
The only down side is that sec^3 is usually quite a labororus integral to calculate :( Hall's hperbolic suggestion is the quickest.
 

Similar threads

High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
3K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Non solvable integral? (dx/dt)^2 dt
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
5K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
2K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Find ##\int_0^π \sin^ n x dx ##
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad How to Approach a Double Exponential Integral?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Challenging integral involving exponentials and logarithms
  • · Replies 14 ·
Replies
14
Views
3K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K