Plot a function with different values of a parameter on the same plot

Question

I want to plot the following function with different values of the parameter $b$:

 a = -0.1 Sqrt[0.001^2 + b^2];

 M[t_] := NIntegrate[10w E^(-w/50)((1-Cos[(w+a)t])/(w+a)^2),{w,0,-a,∞}, 
 MaxRecursion -> 100, AccuracyGoal -> 10, Method -> "PrincipalValue"]

 Plot[M[t],{t,0,1000}]

How can I code different plots for different values of parameter $b$ in a single plot?

Note: My actual code is pretty bigger than this. So I cannot define a new series of variables and functions for each value of the parameter and then plot them simultaneously.

Answer 1

Following J.M.'s comment (his credit)

M[t_, b_] :=  With[{a = -0.1 Sqrt[0.001^2 + b^2]}, 
  NIntegrate[10 w E^(-w/50) ((1 - Cos[(w + a) t])/(w + a)^2), {w, 
    0, -a, ∞}, MaxRecursion -> 100, AccuracyGoal -> 10, 
   Method -> "PrincipalValue"]]

Then

 Plot[ Table[M[t, b], {b, 1, 2}]//Evaluate, {t, 0, 1}]

Answer 2

Check this formalism. It works.

mList = {1, 3, 4, 6};

Plot[Table[m x, {m, mList}] // Evaluate, {x, 0, 10}, 
PlotLegends -> mList]