1. ## Integration

Please am stuck with the integration I kindly need assistance in solving this equation, I will really appreciate if anyone could help with the solution or a guide on what i should do

$\displaystyle \int\limits_0^\infty {\log (1 + \frac{a}{b}x)\frac{{N{e^{ - x}}{x^{y - 1}}}}{{\left( {y - 1} \right)!}}} {\left( {1 - {e^{ - x}}\sum\limits_{i = 0}^{y - 1} {\frac{{{x^i}}}{{i!}}} } \right)^{N - 1}}dx$

2. ## Re: Integration

Hey Igbafe.

Are the y and x variables independent or not?

3. ## Re: Integration

a,b,y and N are sets of constants

4. ## Re: Integration

Mathematica chewed on it for a bit and could not come up with an answer.

5. ## Re: Integration

Are you looking for an estimate or an analytic expression?

6. ## Re: Integration

An analytical expression

7. ## Re: Integration

What is the context? Is it an exercise from an insanely difficult course on analysis, or does it arise during the course of real-world research?

8. ## Re: Integration

it arise during the course of real-world research from a MIMO communication point of view where the best set of Nt out of Nr antennas are selected for transmission and received by Nd antennas. N=combination of(Nr,Nt) while y=(Nt*Nd) hence N and y will be constants depending on wat values of antenna that are available hence am trying to have an analytical expression in terms of N and y

9. ## Re: Integration

My initial suggestion is that you are going to need to either:
c) use linear approximations for the exponentials and logarithm in the integral
d) use numerical methods.

My gut instinct would be to go for d) and write a computer program to do the heavy lifting -- but it would be a project that I expect would take me a number of weeks, months even.

10. ## Re: Integration

updates on some work already done. Basically the equation if of the form

$\displaystyle C(N) = \int\limits_0^\infty {{I_{(N)}}{p_{(N)}}} dx$

where

$\displaystyle {I_{(N)}} = \log \left( {1 + \frac{a}{b}x} \right)$

and

$\displaystyle {p_{(N)}}(x) = \frac{{N\left( {{e^{ - x}}{x^{y - 1}}} \right)}}{{\left( {y - 1} \right)!}}{\left( {1 - {e^{ - x}}\sum\limits_{i = 0}^{y - 1} {\frac{{{x^i}}}{{i!}}} } \right)^{N - 1}}$

we have been able to integrate
$\displaystyle {p_{(N)}}$
within the limits of [0,t] as

$\displaystyle \int_0^t {{p_{(N)}}} (x)dx = \int_0^t {\frac{{N\left( {{e^{ - x}}{x^{y - 1}}} \right)}}{{\left( {y - 1} \right)!}}} {\left( {1 - {e^{ - x}}\sum\limits_{i = 0}^{y - 1} {\frac{{{x^i}}}{{i!}}} } \right)^{N - 1}}dx = \frac{N}{{\left( {y - 1} \right)!}}\sum\limits_{i = 0}^{N - 1} {{{\left( { - 1} \right)}^i}} \left( \begin{array}{c} N - 1\\ i \end{array} \right)\sum\limits_{h = 0}^{i(y - 1)} l \left( {y,\:i} \right)\left[ { - {e^{ - \left( {i + 1} \right)t}}\sum\limits_j^{y + h - 1} {\frac{{j!\left( \begin{array}{c} y + h - 1\\ j \end{array} \right)}}{{{{\left( {i + 1} \right)}^{j + 1}}}}} {t^{y + h - j - 1}} + \frac{{\left( {y + h - 1} \right)!}}{{{{\left( {i + 1} \right)}^{y + h}}}}} \right]$

I don't know if this could help in anyway in solving my problem above

11. ## Re: Integration

Can you get a Taylor series like expansion and cut it off for some error boundary term and integrate the partial series?

12. ## Re: Integration

okay i will appreciate any help on that