The following is given:

Question

Given:

^k∫₀ x e^0,5x dx = 8

Show that:

0,5k -1 = e^0,5k

Answer 1

Use integration by parts where u=x du=dx dv = e^.5x dx and v = 2e^.5x then 8= 2xe^.5x - 2∫₀^ke^.5xdx this gives

8= 2k e^.5k -4e^.5x |₀^k then 8 = 2k e^.5k -4 e^.5k +4 simplifying 4= 2ke^.5k -4e^.5k so 2ke^.5k =4 + 4e^.5k and then

.5ke^.5k =1 + e^.5k divide through by e^.5k .5k = e^-.5k +1 and .5k- 1 = e^-.5k