∫ 0 ∞ x n e − a x b d x = 1 b a − n + 1 b Γ [ ( n + 1 ) / b ]Is there a way to evaluate this integral with a lower limit K&gt;0?

Question

∫    0    ∞        x    n        e          −      a              x        b              d  x  =      1    b        a          −                        n          +          1                b              Γ  [  (  n  +  1  )      /    b  ]Is there a way to evaluate this integral with a lower limit K&amp;gt;0?

Yasminru · Accepted Answer

The definition of the incomplete Gamma function is  Γ  (  s  ,  z  )  =      ∫    s    ∞        x          z      −      1            e          −      x        d  xUsing the substitution   t  =  a      x    b   gets you the same result in the post, integrand, prefactor and all. The only thing that changes is the lower bound goes from   K  ↦  a      K    b   which means the answer is just simply      ∫    0    ∞        x    n        e          −      a              x        b              d  x  =      1    b        a          −                        n          +          1                b              Γ      (    a          K      b        ,                  n        +        1            b        )  Fun fact, all I did was copy paste your equation and inserted the new argument.

∫^∞_0 x^n e^−ax^b dx=1/b a^−n+1/bΓ[(n+1)/b] Is there a way to evaluate this integral with a lower limit K>0?

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