Understanding the Value of the Exponential Integral in Advanced Calculus

The mathematical problem of evaluating the integral of the form (int e^x frac{x-3}{(x-1)^3}dx) is a classic example of a problem in advanced calculus. This integral challenges one's understanding of integration techniques and highlights the importance of manipulating expressions to simplify the integration process.

Evaluating the Integral

Let's first clarify a small typo in the given integral. The correct form should be:

[ int e^x frac{x-3}{(x-1)^3}dx ]

Breaking down the integral, we can rewrite the numerator:

[ int e^x frac{x-3}{(x-1)^3}dx int e^x frac{x-1-2}{(x-1)^3}dx int e^x left(frac{1}{(x-1)^2}-frac{2}{(x-1)^3}right)dx ]

This simplification allows us to split the integral into two parts:

[ int e^x left(frac{1}{(x-1)^2}-frac{2}{(x-1)^3}right)dx int e^x left(frac{1}{(x-1)^2}right)dx - 2int e^x left(frac{1}{(x-1)^3}right)dx ]

Applying Integration Techniques

Now, let's consider the functions:

$f(x) frac{1}{(x-1)^2}$ $g(x) frac{1}{(x-1)^3}$

Both these functions are candidates for integration by parts or substitution. However, a more straightforward approach is to recognize the form of the integrals and apply known properties of the exponential integral.

Integration by Parts

For the first integral, we can use integration by parts where:

$u frac{1}{(x-1)^2}$ $dv e^x dx$

This gives:

$du -frac{2}{(x-1)^3}dx$, $v e^x$

Using the formula for integration by parts, $uv - int v du$, we get:

$int e^x frac{1}{(x-1)^2} dx e^x frac{1}{(x-1)^2} 2int e^x frac{1}{(x-1)^3} dx$

Final Integration

Combining the results, we get:

$int e^x left(frac{1}{(x-1)^2} - frac{2}{(x-1)^3}right) dx e^x frac{1}{(x-1)^2} C$

where $C$ is the constant of integration.

What is the Exponential Integral?

The exponential integral, often denoted as $Ei(x)$, is a special function used in mathematics and physics. It is defined by the integral:

$Ei(x) -int_{-x}^{infty} frac{e^{-t}}{t} dt$

This function is particularly useful in solving differential equations and in the study of particle interactions in physics. Understanding the exponential integral is crucial for advanced calculus and its applications in various scientific fields.

Related Sections:

Wikipedia: Exponential Integral Math StackExchange: Pulling the Exponential Function Out of an Integral Sign Calculus Tutorial: Integration by Parts

Thank you for reading. We look forward to seeing you again for more insightful content on advanced calculus and its applications.