Solve Integral Analytically: Int[(1+x^2)^-n e^-x^2]dx

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    Analytical Integral
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SUMMARY

The integral of the form Int[(1+x^2)^-n e^-x^2]dx can be approached analytically by recognizing that the expression can be rewritten as (1+x^2)^n e^-x^2. The discussion confirms that while the infinite integral can be computed, the indefinite integral is expressible solely in terms of the error function (erf). This method provides a clear pathway for solving similar integrals analytically.

PREREQUISITES
  • Understanding of integral calculus and the properties of integrals.
  • Familiarity with the error function (erf) and its applications.
  • Knowledge of exponential functions and their behavior in integrals.
  • Basic algebraic manipulation skills, particularly with powers and reciprocals.
NEXT STEPS
  • Study the properties and applications of the error function (erf).
  • Learn techniques for solving integrals involving exponential functions.
  • Explore methods for evaluating infinite integrals analytically.
  • Investigate advanced integral calculus techniques, including substitution and integration by parts.
USEFUL FOR

Mathematicians, students of calculus, and anyone interested in advanced integral techniques will benefit from this discussion, particularly those working with integrals involving exponential decay and polynomial expressions.

matt_crouch
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can someone suggest a method to solve an integral of the form analytically?
##\int \left[\frac{1}{(1+x^{2})}\right]^{-n}e^{-x^{2}}dx##
 
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Is the bracket a floor function? Otherwise you can use the reciprocal to get (1+x2)n. In that case the infinite integral can be done, but the indefinite integral can be expressed only in terms of the erf.
 

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