# Decay formula with Improper Integrals

1. Sep 27, 2008

### lelandsthename

1. The problem statement, all variables and given/known data
Hey everyone! I have another question about improper integrals, they're so hard!

M = -k $$\int$$$$\stackrel{inf.}{0}$$ te^(kt) dt

When k = -0.000121 (Carbon 14's constant, we are solving for the mean life of a carbon-14 isotope)

2. Relevant equations
Improper integrals, integration by parts

3. The attempt at a solution

=- k lim$$_{t->inf.}$$ $$\int$$te^(kt) dt from 0 to infinity

by parts:
u = t
du = dt
dv = e^(k) dt
v = (1/k)e^(-kt)

=(t((1/k)e^(kt)) + $$\int$$(1/k)e^(kt)dt

=((t)/(k))e^(-0.000121t) - (1/((k)^2)e^(kt)

=- k lim$$_{t->inf.}$$ $$\int$$(t)/(k)e^(kt) - (1/(k)^2)e^(kt)

Where can I go from here? I can put both terms over (k)^2 but the limit of that term times e^(k) equals 1*, right? So is hte answer just -k? Somehow I am skeptical!

Last edited: Sep 27, 2008
2. Sep 27, 2008

### tiny-tim

Hi lelandsthename!

oooh … you've made this so complicated!

why not just leave k as k until the very end?
Nooo … e-∞ = 0, but e0 = 1.

3. Sep 27, 2008

### lelandsthename

haha! ok I will give it a try with k first, it'll even make it look neater! and thank you for helping me with the limit! So because the limit equals one then the answer will just be -k?