Numerical Evaluation of Some Definite Integrals

f(x)

log(x)(1+100

2

x

)

numerical evaluation

numerical minus analytic

x

0.836

This Demonstration shows a trick for computing the definite integral

F(x)=

x

∫

0

f(x)dx

numerically in a given interval of its upper bound

x

using Mathematica. Instead of using NIntegrate we use the function NDSolve. Five test functions are borrowed from reference[1]. Four of these test functions have a singular point at

x=0

. You can plot the analytic solutions of the test integrals as well as the difference of the numerical and analytic solutions as a function of the upper bound variable

x

for different working precisions. The black point on the curve gives the function value

F(x)

when the slider position is at

x

. At the right endpoint these values coincide with the values given in the reference in the details.

Details

[1] IMSL Fortran Library User's Guide, MATH/LIBRARY Volume 1 of 2, http://mesonpi.cat.cbpf.br/ssolar/imsl.htm.

External Links

Definite Integral (Wolfram MathWorld)

NDSolve (Wolfram Documentation Center)

NIntegrate (Wolfram Documentation Center)

Permanent Citation

Mikhail Dimitrov Mikhailov

"Numerical Evaluation of Some Definite Integrals"
http://demonstrations.wolfram.com/NumericalEvaluationOfSomeDefiniteIntegrals/
Wolfram Demonstrations Project
Published: March 7, 2011