Indefinite integral for the square root of a quartic function

Click For Summary
SUMMARY

The discussion focuses on the challenge of integrating the square root of a quartic function represented as ((At4 + Bt3 + Ct2 + Dt + E)1/2) dt. Participants note that traditional reduction formulas typically apply to quadratic functions, making direct integration of quartics complex. The integral is confirmed to involve inverse sine functions and elliptic integrals of both the first and third kinds, as indicated by Wolfram Alpha's analysis. The integration process requires specific constants A, B, C, D, and E to be defined post-integration.

PREREQUISITES
  • Understanding of quartic functions and their properties
  • Familiarity with integral calculus and integration techniques
  • Knowledge of elliptic integrals and their applications
  • Experience using computational tools like Wolfram Alpha for complex integrals
NEXT STEPS
  • Study the properties and applications of elliptic integrals of the first and third kinds
  • Learn techniques for converting quartic functions into solvable forms
  • Explore advanced integration techniques in calculus, particularly for non-standard forms
  • Investigate numerical methods for approximating integrals of complex functions
USEFUL FOR

Mathematicians, calculus students, and anyone involved in advanced mathematical modeling or computational analysis of quartic functions.

StTheo
Messages
2
Reaction score
0
I arrived at this problem while trying to find the length of a 3D http://en.wikipedia.org/wiki/Cubic_interpolation" .

Basically, I'm having to figure out how to integrate ((At4+Bt3+Ct2+Dt+E)1/2)dt

A,B,C,D, and E are constants which have to be plugged in after integration, I'm sorry to say.

All the reduction formulas I've seen use quadratic functions of some kind, so would I have to convert the quartic function into some kind of quadratic function?
 
Last edited by a moderator:
Physics news on Phys.org
In general, it cannot be done simply, I believe. From what I can tell using Wolfram-Alpha, the integral involves the inverse sine function as well as the elliptic integrals of the first AND third kinds. It can't evaluate the integral, though: here.
 
Grr, oh well. However, I hadn't seen that wonderful site you linked to before. I'll probably waste quite a bit of time on it.
 

Similar threads

Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
2K
Graduate Need help with an integral -- How to integrate velocity squared?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Integral of a sinc squared function over a square root function
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Memory issues in numerical integration of oscillatory function
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why is the integral of ##\arcsin(\sin(x))## so divisive?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Integrating a function of which poles appear on the branch cut
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Square of an integral containing a Green's Function
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad What _is_ an indefinite integral?
  • · Replies 12 ·
Replies
12
Views
3K
Graduate Can this difficult Gaussian integral be done analytically?
  • · Replies 5 ·
Replies
5
Views
2K