Decay formula with Improper Integrals

[lelandsthename]

Homework Statement

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)

Homework Equations

Improper integrals, integration by parts

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!

 

[tiny-tim]

Hi lelandsthename!

oooh … you've made this so complicated!

why not just leave k as k until the very end?

… I can put both terms over (0.000121)^2 but the limit of that term times e^(-0.000121t) equals 0, right?

Nooo … e^-∞ = 0, but e⁰ = 1.

 

[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?