What is #f(x) = int xe^(2-x) + 3x^2 dx# if #f(0 ) = 1 #?
Begin by using the sum rule for integrals and splitting these into two separate integrals:
The first of these mini-integrals is solved using integration by parts:
Now using the integration by parts formula
The second of these is a case of the reverse power rule, which states:
We are given the initial condition
Making this final substitution, we obtain our final solution: