Why am I getting a negative value from $\int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$?

This is a precision issue.

$Version

(* "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)" *)

Since you used a machine precision number (i.e., 0.5) in the integrand, the integration is done with only machine precision.

Integrate[E^(-16/x^8^(-1) - (2*x)/(0.5)^9)/x^(9/8), {x, 0, Infinity}]

(* -0.0824979 *)

Using exact numbers

int = Integrate[E^(-16/x^8^(-1) - (2*x)/(1/2)^9)/x^(9/8), {x, 0, Infinity}]

(* (1/(2^(3/4) \[Pi]^(
 7/2)))MeijerG[{{}, {}}, {{-(1/8), 0, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/
   8}, {}}, 262144] *)

If this complicated result is converted to a number using machine precision

int // N

(* 0.0270485 *)

However, using arbitrary-precision

int // N[#, 20] &

(* 5.6160550672798812373*10^-16 *)

Comparing with numerical integration

int = NIntegrate[E^(-16/x^8^(-1) - (2*x)/(1/2)^9)/x^(9/8), {x, 0, Infinity}]

(* 5.61606*10^-16 *)

This value is consistent with the earlier arbitrary-precision result and does not change with increased precision.

int = NIntegrate[E^(-16/x^8^(-1) - (2*x)/(1/2)^9)/x^(9/8), {x, 0, Infinity}, 
  WorkingPrecision -> 20]

(* 5.6160550672798812757*10^-16 *)

Looking at the integrand

max = NMaximize[
  {E^(-16/x^8^(-1) - (2*x)/(1/2)^9)/x^(9/8),
   x > 0}, x, WorkingPrecision -> 20]

(* {1.3826054145707193900*10^-13, {x -> 0.0029477253331967126781}} *)

LogPlot[E^(-16/x^8^(-1) - (2*x)/(1/2)^9)/x^(9/8), {x, 1*^-4, 2*^-2}, 
 WorkingPrecision -> 20,
 Epilog -> {Red, AbsolutePointSize[4],
   Point[{x /. max[[2]], Log@max[[1]]}]}]

enter image description here

Consequently, a small value for the integration is expected.


Another case of using non-exact numbers with exact function.

See the difference:

integrand = x^(-9/8) Exp[-16 x^(-1/8) - 2 x/(1/2)^9];
Integrate[integrand, {x, 0, Infinity}] // N

Mathematica graphics

integrand = x^(-9/8) Exp[-16 x^(-1/8) - 2 x/(0.5)^9];
Integrate[integrand, {x, 0, Infinity}]

Mathematica graphics

Rule of thumb: use exact numbers when calling exact functions like DSolve, Integrate, etc.. as sometimes when using non-exact numbers, Mathematica result can go wrong.

Tags:

Numerical Integration

Calculus And Analysis

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