INTEGRAL question...

Question

INTEGRAL (from 0 to infinity) of ( x^c / c^x ) dx

Who knows the answer of this integral in DESCRIPTIVE? I need the answer STEP BY STEP to the last part o answer. Please help me, thats very important to me.

(My teacher said in case of finding the answer easier we should suppose:

c^x = e^(xlnc)

xlnc=t

).

Thanks all again. :)

Live in the best way! :)

Answer 1

Use the substitutiton.

x = t/ln c

dt = lnc dx

dt/ln c = dx

1/ln(c) int(0 to infinity) [t/ln(c) * e^t dt]

1/ln^2(c) int(0 to infintiy)[t/e^t dt]

You can do the integral by parts. Let u = t, dv = e^(-t) dt, du = dt, v = -e^(-t)

int(u dv) = uv - int(v du)

-te^(-t)+int(e^(-t)dt) = -te^(-t) -e^(-t) evaluated from 0 to infinity

At infinity, e^(-t) goes to zero. So it's 0-(-1) = 1.