Integrating exponentials

[Chocolate Muffins And Pie]

I keep getting frustrated with problems requiring integration of an exponential. This has been helpful, feel free to point out other ways or poke holes in this.

To integrate x^a*e^-bx

1) write out -e^-bx* ( )
2) in the parentheses, start with x^a and add a term for each derivative until you reach a constant (so x^a + ax^(a-1) +a(a-1)x^(a-2).....)
3) for each of those terms, add a denominator starting with b^1 and increasing the exponent until you reach b^(a+1)
4) evaluate at the limits of the integral.

[asile]

Originally Posted by Chocolate Muffins And Pie

I keep getting frustrated with problems requiring integration of an exponential. This has been helpful, feel free to point out other ways or poke holes in this.

To integrate x^a*e^-bx

1) write out -e^-bx* ( )
2) in the parentheses, start with x^a and add a term for each derivative until you reach a constant (so x^a + ax^(a-1) +a(a-1)x^(a-2).....)
3) for each of those terms, add a denominator starting with b^1 and increasing the exponent until you reach b^(a+1)
4) evaluate at the limits of the integral.

Why not just use the fact that this is a constant away from being a gamma distribution? It integrates to 1 / constant if you are going from 0 to infinity.

Otherwise, it might be faster to back out the constant and use the gamma -> poisson trick for an integer n/alpha parameter.