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I have some problem to integrate the expression given below:

hN = 5; γ = 30; ρ = 1200; cp = 1200; k = 0.2; 
w0 = 150*10^-6; α = k/(ρ*cp); τ = 0; 
ab = hN/(hN - γ*k)*Exp[hN^2/k*α (t - τ)]* Erfc[hN/k*Sqrt[α (t - τ)]];
ac = (γ*k)/(hN - γ*k)* Exp[γ^2*α (t - τ)]* Erfc[γ*Sqrt[α (t - τ)]]; 
cd = (ab - ac);

Integrate[
  Exp[-2*r^2/8*α*(t - τ) + w0^2]/(8*α*(t - τ) + w0^2)*cd, 
  {t, 0, .001}
]
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  • $\begingroup$ You haven't set a value for r. Are you trying to get the integral as a function of r? Setting r = 1 and using NIntegrate (Integrate can't cope, so do it numerically) seems to evaluate fine. $\endgroup$ – aardvark2012 Aug 21 '17 at 12:01
  • $\begingroup$ Yes ,I want to integral in term of r. $\endgroup$ – Gopal Verma Aug 21 '17 at 12:42
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    $\begingroup$ So, why was there a period in the title, Gopal? $\endgroup$ – J. M. will be back soon Aug 21 '17 at 14:59
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    $\begingroup$ This is an almost constant function of r. Here are some values obtained by numerical integration:$0.5 \to 43360.85509867566,\, 0. \to 43360.85509886233,\, 20 \to 43360.85480019366.$ $\endgroup$ – user64494 Aug 21 '17 at 16:56
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    $\begingroup$ @user64494 Not really. You just didn't span a large enough value of $r$ to see an appreciable change. $\endgroup$ – MarcoB Aug 21 '17 at 20:28
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   Finally, I got our results that I want.
         hN = 5; \[Gamma] = 30; \[Rho] = 1200; cp = 1200; k = 0.2;
        w0 = 150*10^-6; \[Alpha] = k/(\[Rho]*cp); \[Tau] = 0;
         ab = hN/(hN - \[Gamma]*k)*Exp[hN^2/k*\[Alpha] (t - \[Tau])]*
       Erfc[hN/k*Sqrt[\[Alpha] (t - \[Tau])]];
        ac = (\[Gamma]*k)/(hN - \[Gamma]*k)*
       Exp[\[Gamma]^2*\[Alpha] (t - \[Tau])]*
       Erfc[\[Gamma]*Sqrt[\[Alpha] (t - \[Tau])]];
         cd = (ab - ac);
        int[r_?NumericQ, u_?NumericQ] := 
        NIntegrate[
       Exp[-2*r^2/8*\[Alpha]*(t - \[Tau]) + 
      w0^2]/(8*\[Alpha]*(t - \[Tau]) + w0^2) cd, {t, 0, u}]
      Plot[Evaluate[Table[int[r, u], {u, 0.04, 1, .2}]], {r, -10^5, 10^5}, 
       PlotRange -> All]

enter image description here

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Symbolic integration seems to take forever, and a solution is not guaranteed. If you can make do with numerical values, however, numerical integration is very fast:

Clear[int]
int[r_?NumericQ] := NIntegrate[
  Exp[-2*r^2/8*α*(t - τ) + w0^2]/(8*α*(t - τ) + w0^2)*cd, {t, 0, .001}]

Plot[int[x], {x, -10^6, 10^6}]

Mathematica graphics

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  • $\begingroup$ Thanks, Actually I want to make a table of graph ( e.g. t=(0,.001,.002,..) in same graph and x-axis range. $\endgroup$ – Gopal Verma Aug 21 '17 at 22:17
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    $\begingroup$ @GopalVerma I'm not sure I understand. You want to plot the value of the integral as you change the limits of integration? For a constant value of $r$? Or you want a 3D plot, as a function of $t_{max}$ and $r$? $\endgroup$ – MarcoB Aug 21 '17 at 22:51
  • $\begingroup$ Yes, I want to plot the value of integral by changing the limit, but in same graph,I want to plot it for compaion. $\endgroup$ – Gopal Verma Aug 22 '17 at 8:16

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