Integration of product of two exponential functions

May 2010
1
0
Hello,

I apologise in advance if this is a naive question (which it probably is) but my maths is very rusty and my old notes and text books don't seem to cover this particular problem.

What is the correct approach to integrating the product of two exponential functions of the form below?

I have current, i(t) = 10e^-5000t

voltage, v(t) = 50(1-e^-5000t).

Power, p(t) = i(t).v(t)

Energy for t is greater than equal to zero is therefore the integral of p(t) for t = 0 to infinity.

I know that the answer is 50 mJ (or 0.050 joules) from calculating p(t) in a spreadsheet for small increments of t until p(t) tends to zero but cannot demonstrate it mathematically.

I've tried integrating by parts but this does not seem to close the deal as I always end up with a integral of the same or similar form to evaluate.

What is the blindingly obvious thing that I am missing ? Am i on the right track with integrating by parts or is there an easier reduction of the problem given the similarities between the two functions?

Many thanks
 
Oct 2009
4,261
1,836
Hello,

I apologise in advance if this is a naive question (which it probably is) but my maths is very rusty and my old notes and text books don't seem to cover this particular problem.

What is the correct approach to integrating the product of two exponential functions of the form below?

I have current, i(t) = 10e^-5000t

voltage, v(t) = 50(1-e^-5000t).

Power, p(t) = i(t).v(t)

Energy for t is greater than equal to zero is therefore the integral of p(t) for t = 0 to infinity.

I know that the answer is 50 mJ (or 0.050 joules) from calculating p(t) in a spreadsheet for small increments of t until p(t) tends to zero but cannot demonstrate it mathematically.

I've tried integrating by parts but this does not seem to close the deal as I always end up with a integral of the same or similar form to evaluate.

What is the blindingly obvious thing that I am missing ? Am i on the right track with integrating by parts or is there an easier reduction of the problem given the similarities between the two functions?

Many thanks

According to what you wrote, you have the improper integral:

\(\displaystyle \int\limits^\infty_010e^{-5,000t}\cdot 50(1-e^{-5,000t})\,dt=\) \(\displaystyle 500\int\limits_0^\infty\left(e^{-5,000t}-e^{-10,000t}\right)\,dt=\) \(\displaystyle 500\lim_{b\to\infty}\left[-\frac{1}{5,000}\,e^{-5,000t}+\frac{1}{10,000}\,e^{-10,000t}\right]^b_0=\) \(\displaystyle \frac{1}{20}\lim_{b\to\infty}\left(-\frac{2}{e^{5,000b}}+2e^0+\frac{1}{e^{10,000b}}-e^0\right)=\)

\(\displaystyle =\frac{1}{20}=0.05\)

Tonio
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
Multiply the functions together before integrating, what do you get?