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Mar 30th, 2015
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  1. Integrate[Exp[-y x^α1 ], {x, 0, x0}, Assumptions -> {1 < α1, x0 > 0, y > y0, y0 > 0}] + Integrate[Exp[-x0^(α1 - α2) y x^α2], {x, x0, Infinity}, Assumptions -> {1 < α1, 0 < α2 < 1, x0 > 0, y > y0, y0 > 0, x0 > 0}]
  2.  
  3. (x0 ExpIntegralE[(-1 + α2)/α2, x0^α1 y])/α2 +(y^(-1/α1) (Gamma[1/α1] -Gamma[1/α1, x0^α1 y]))/α1
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  5. Integrate[(x0 ExpIntegralE[(-1 + α2)/α2, x0^α1 y])/α2 + (y^(-1/α1) (Gamma[1/α1] - Gamma[1/α1, x0^α1 y]))/α1, {y, y0, Infinity}, Assumptions -> {α1 > 1, 1 > α2 > 0, x0 > 1, y0 > 0}]
  6.  
  7. (y0^(1 - 1/α1) Gamma[1/α1])/(1 - α1) + (y0^((-1 + α1)/α1) Gamma[1/α1, x0^α1 y0])/(-1 + α1) + ( x0 ((E^(-x0^α1 y0)x0^-α1 (α1 - α2))/(-1 + α1) - x0^(-(α1/α2)) y0^((-1 + α2)/α2) Gamma[1/α2, x0^α1 y0]))/(-1 + α2)
  8.  
  9. With[{α1 = 1.4, α2 = 0.8, x0 = 8, y0 = 1}, ( y0^(1 - 1/α1) Gamma[1/α1])/(1 - α1) + ( y0^((-1 + α1)/α1) Gamma[1/α1, x0^α1 y0])/(-1 + α1) + ( x0 ((E^(-x0^α1 y0) x0^-α1 (α1 - α2))/(-1 + α1) - x0^(-(α1/α2)) y0^((-1 + α2)/α2) Gamma[1/α2, x0^α1 y0]))/(-1 + α2)]
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  11. With[{α1 = 1.4, α2 = 0.8, x0 = 8, R = 10, y0 = 1}, NIntegrate[ Piecewise[{{Exp[-y x^α1 ], x <= x0}, {Exp[-x0^(α1 - α2) y x^α2], x > x0}}], {x, 0, Infinity}, {y, y0, Infinity}]]
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