# Really complex function to integrate

• womfalcs3
In summary, Ice is trying to find a software application that can handle a very complex function, and has difficulty doing it by hand. She is looking for a software application that can approximate the exponential function.
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.

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.

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

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

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

Last edited:

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.

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-y1/7)2

?

## What is a really complex function to integrate?

A really complex function to integrate is a mathematical expression that involves multiple variables, trigonometric functions, and other complex operations that make it difficult to find an exact solution. These functions often require advanced techniques and tools to solve.

## Why is it important to integrate complex functions?

Integrating complex functions is crucial in many areas of science, including physics, engineering, and economics. It allows us to calculate important quantities such as areas, volumes, and rates of change, which are essential for understanding and predicting real-world phenomena.

## What are some common challenges in integrating complex functions?

Some common challenges in integrating complex functions include dealing with multiple variables, determining appropriate integration limits, and selecting the right integration technique. These challenges can make the integration process time-consuming and require a deep understanding of mathematical concepts.

## What techniques can be used to integrate complex functions?

There are several techniques that can be used to integrate complex functions, including substitution, integration by parts, trigonometric substitution, and partial fractions. These techniques often involve breaking down the function into simpler parts and using rules and formulas to solve them.

## How can one improve their skills in integrating complex functions?

Improving skills in integrating complex functions requires practice and a strong foundation in mathematics. It is also helpful to familiarize oneself with different integration techniques and their applications. Additionally, using software and online tools can aid in checking solutions and understanding the steps involved in integrating complex functions.

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