Integration help please?

Question

How do you do the following:

1) ∫ x² * e^(px) dx
2) ∫ x² * e^(-px) dx

Note that p and x are both variables here.

Answer 1

1)
For integration, p is a constant. It may vary as a parameter.
∫ x² * e^(px) dx
= x^2 ∫ e^(px) dx - ∫ [ d/dx(x^2) * ∫ e^(px) dx ] dx
= (1/p) x^2 * e^(px) - (2/p) ∫ x * e^(px) dx
= (1/p) x^2 * e^(px) - (2/p) [ x ∫ e^(px) dx - ∫∫ [e^(px) dx] dx]
= (1/p) x^2 * e^(px) - (2/p) [ (x/p) e^(px) + (1/p^2)e^(px) ] + c
= (1 / p^3) * e^(px) [ p^2 x^2 - 2px + 2 ] + c

2)
Replacing p by - p
∫ x² * e^(-px) dx = ( - 1/p^3) * e^(-px) [ p^2 x^2 + 2px + 2] +c

Answer 2

p may be a variable, but unless you do a double integration with dp, it is treated as a constant. These are standard forms and can be looked up in a table of integrals.

Answer 3

With respect to the integration "p" is a constant , termed a parameter in this problem. Use integration by parts