Convergence of ∫dx/sqrt(x^4+1): Explanation Needed

  • Thread starter Thread starter MozAngeles
  • Start date Start date
  • Tags Tags
    Convergence
Click For Summary
SUMMARY

The integral ∫dx/sqrt(x^4+1) from x=-∞ to x=∞ converges. The solution involves applying the limit comparison test and direct comparison to analyze the behavior of the integrand as x approaches ±∞. By breaking the integral into two parts, ∫dx/√(x^4+1) from x=-∞ to x=0 and ∫dx/√(x^4+1) from x=0 to x=∞, one can effectively evaluate convergence. The comparison test confirms that the integral converges due to the rapid growth of the denominator.

PREREQUISITES
  • Understanding of improper integrals
  • Familiarity with limit comparison test
  • Knowledge of direct comparison test
  • Basic calculus concepts involving integrals
NEXT STEPS
  • Study the limit comparison test in detail
  • Explore direct comparison test applications
  • Learn about convergence criteria for improper integrals
  • Investigate similar integrals and their convergence properties
USEFUL FOR

Students studying calculus, particularly those focusing on improper integrals and convergence tests, as well as educators seeking to explain these concepts effectively.

MozAngeles
Messages
100
Reaction score
0

Homework Statement


Does ∫dx/sqrt(x^4+1) from x=-∞ to x=∞ converge or diverge?
explain in detail if you can please.
thanks




Homework Equations


limit comparison test
direct comparison


The Attempt at a Solution

...well i have the answer, it converges. I just need a better explanation than the solutions manual gives.

the first thing i did was break it up into ... ∫dx/√(x4+1) from x=-∞ to x=0 +∫dx/√(x4+1) from x=0 to x=∞ then i was stuck, now I'm here...
 
Physics news on Phys.org
i think a comparison test would work well here...
 

Similar threads

Prove limit comparison test for Integrals
  • · Replies 2 ·
Replies
2
Views
2K
Checking convergence of Gaussian integrals
  • · Replies 47 ·
2
Replies
47
Views
5K
Finding if an improper integral is Convergent
  • · Replies 5 ·
Replies
5
Views
2K
The integral of (sin x + arctan x)/x^2 diverges over (0,∞)
  • · Replies 5 ·
Replies
5
Views
3K
Integral of 1/ln(x). Convergence test
  • · Replies 3 ·
Replies
3
Views
7K
Integral Convergence: Examining 1/(e^x sqrt(x))
  • · Replies 9 ·
Replies
9
Views
3K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Question on Two Differentiation Techniques
  • · Replies 25 ·
Replies
25
Views
2K
Intervals of Convergence- Power Series
  • · Replies 11 ·
Replies
11
Views
3K
How Do You Solve the Differential Equation dy/dx = 1 - y^2?
  • · Replies 12 ·
Replies
12
Views
3K