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How can I solve this integration? I want to solve this integration with c, B, H, Y, b, w surviving as constants in the result:

ClearAll[a, b, t, R1, R2 c, t, H, Y, w]
syms = {c, B, H, Y, b, w}
a = Exp[((t)^(c) - 1)]

R1 = Integrate[a^(-3*w - 3), t]

Second question

a = Exp[(1/\[Beta]) *((t^\[Beta]) - 1)];
R1 = a*(2/(3*w))*
  Exp[(2/(3*w))*Integrate[Exp[-m*t^k], t] /. {m -> 3*(w + 1)/\[Beta], 
     k -> \[Beta]}]

f = D[R1, t]

p1 = Plot[f /. {\[Beta] -> .58, w -> -1.1}, {t, 0, 10}, 
  AxesOrigin -> {0, 0}]`
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    $\begingroup$ I don't think there is anything wrong with your code. Mathematica can't do the integral. However, it can do it for specific integer values of c. Does that help? $\endgroup$
    – mikado
    Dec 2, 2018 at 20:26

1 Answer 1

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It is necessary to use the well-known integral

Integrate[Exp[-m*t^k], t]

(*Out[]= -((t (m t^k)^(-1/k) Gamma[1/k, m t^k])/k)*)

Let's transform

a = Exp[((t)^(c) - 1)];(* a^(-3*w - 3)=Exp[3*w+3]*Exp[-3*(w+1)*t^c]*);

Consequently

R1=Exp[3 + 3*w]*Integrate[Exp[-m*t^k], t] /. {m -> 3*(w + 1), k -> c}
(*Out[]= -((3^(-1/c) E^(3 + 3 w) t (t^c (1 + w))^(-1/c)
   Gamma[1/c, 3 t^c (1 + w)])/c)*)

Let's check that the derivative D[R1,t]coincides with the original function.

R1 = -((3^(-1/c) E^(3 + 3 w) t (t^c (1 + w))^(-1/c) Gamma[1/c, 
        3 t^c (1 + w)])/c);
f = D[R1, t]
(*Out[]=E^(3 + 3 w - 3 t^c (1 + w)) + (
 3^(-1/c) E^(3 + 3 w) t^c (1 + w) (t^c (1 + w))^(-1 - 1/c)
   Gamma[1/c, 3 t^c (1 + w)])/c - (
 3^(-1/c) E^(3 + 3 w) (t^c (1 + w))^(-1/c) Gamma[1/c, 3 t^c (1 + w)])/
 c*)
{p1 = Plot[f /. {c -> .58, w -> -1.1}, {t, 0, 10}, 
   AxesOrigin -> {0, 0}], 
 p2 = Plot[a^(-3*w - 3) /. {c -> .58, w -> -1.1}, {t, 0, 10}, 
   PlotStyle -> Orange, AxesOrigin -> {0, 0}], Show[p1, p2]}

fig1

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  • $\begingroup$ thank you, how can ı get the differentiation of R1 with m, c surviving as constants in the result $\endgroup$
    – merve
    Dec 3, 2018 at 8:46
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    $\begingroup$ @merve Do you want to calculate the derivative D[R1,c]? $\endgroup$ Dec 3, 2018 at 12:10
  • $\begingroup$ not actually D[R1,c], I want to calculate D[R1,t] and then plot R1 versus t (with the range of 0-10). with c= 0.58, w = -1.1 values but I couldn't. $\endgroup$
    – merve
    Dec 3, 2018 at 12:21
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    $\begingroup$ I updated the code added pictures $\endgroup$ Dec 3, 2018 at 13:46
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    $\begingroup$ This is due to the fact that the function f[t] is complex, so use p1 = Plot[Re[f /. {\[Beta] -> .58, w -> -1.1}], {t, 0, 10}, AxesOrigin -> {0, 0}] $\endgroup$ Dec 3, 2018 at 18:21

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