Really complex function to integrate

[womfalcs3]

Hello,

I have a really complex function to integrate (not homework), and I was wondering if there is any software application that can handle it.

I have tried MathCAD and Maple, but both can't perform it. I don't think I can do it by hand, with all the integration by parts and expansions I would have to go through.

I would also have to go at it term by term once expanded. (The numerator expands to 5 terms.)

Here is the integral:

u and delta are functions of x, and P, Ti, and Tinf are constants (I could take out those that can be taken out of the integrand, but it doesn't get me anywhere with the process.).


So I need a symbolic integration.

 

[womfalcs3]

I've worked to simplify the integral to this... taking out constants, and using an very good approximation for the exponential function below. I've also changed the interval, so that the approximation is valid.

I don't think I can do it numerically, because I can't use quadruture symbolically. I would have to plug it many data values and then fit a curve to the plot. I won't do that.

Capital delta is "Ti-Tinf".

The numerator is one of 5 terms I have to integrate one by one as part of the expansion.

 

[ice109]

$$\int 1.38346 P ~u ~e^{-25 y/8}~ \left(1-\frac{y}{8}\right)^3~ \sqrt[7]{y}~ \text{dy} =$$

$$-\frac{0.00270207 P u \left(0.32 e^{3.125 y} (y-9.68055) y \left(y^2-15.3252 y+60.7908\right)+22.8611 (-y)^{6/7} \Gamma<br /> (0.142857,-3.125 y)\right)}{y^{6/7}}$$

on the other hand with a sum in the denominator not mathematica doesn't integrate it

 

[womfalcs3]

Thanks Ice, but the symbol isn't an "8", but rather a small-case delta, which is a function of x. Also, the denominator is imperative, because I'm integrating the product of velocity and the ideal gas law set equal to density.

Density = P/(R*T)

So I need the denominator.

 

[womfalcs3]

I'm thinking I have to approximate the expenonetial function using a polynomial for the small interval. Since I need an analytial method of solution.

I already have a linear function that can be within 20% error.


Is there a good polynomial expression (Where the power is 1 or larger for all powers in the function.) to approximate:

f(y)=(1-y^1/7)²

?