Integral of x+1 / x-b dx

Introduction:

In the realm of calculus, one of the fundamental operations is integration, which seeks to find the antiderivative of a given function. While basic integrals may be relatively straightforward, certain expressions require more intricate techniques for resolution. In this article, we will delve into the integral of $\frac{�+1}{�-�}$, where $�$ is a constant. We will explore the process of solving this integral using polynomial division, shedding light on the importance of such methods in handling complex mathematical problems.

Understanding Rational Functions:

A rational function is defined as a ratio of two polynomials. In this context, our rational function is $\frac{�+1}{�-�}$, where both the numerator and denominator are polynomials. When integrating such functions, we often encounter the need for algebraic manipulation to simplify and find a suitable antiderivative.

The Integral in Question:

Our integral is:

$\int \frac{�+1}{�-�}��$

To solve this integral, we can employ polynomial division, a technique that allows us to divide the numerator by the denominator to express the integrand as a sum of simpler terms.

Polynomial Division:

Let's perform polynomial division with $�-�$ dividing $�+1$:

$�-�\mathrm{\mid }�+1$ $-(�-�)$ __________ $1+�$

The result is $\frac{1+�}{�-�}$. Our integral now becomes:

$\int \frac{�+1}{�-�}��=\int (1+\frac{1+�}{�-�})��$

Breaking Down the Integral:

We can now break down the integral into two parts:

1. The integral of $1$ with respect to $�$, which is simply $�$.

2. The integral of $\frac{1+�}{�-�}$ with respect to $�$.

Integral of $1$:

The integral of the constant $1$ with respect to $�$ is straightforward:

$\int 1��=�+{�}_1$

Here, ${�}_1$ represents the constant of integration.

Integral of $\frac{1+�}{�-�}$:

Now, let's focus on the second part of the integral, which involves $\frac{1+�}{�-�}$. This is an example of a logarithmic integral. The integral of $\frac{1}{�-�}$ is the natural logarithm of the absolute value of $�-�$, and the presence of $1+�$ as a constant multiplier does not affect this result:

$\int \frac{1+�}{�-�}��=(1+�)\int \frac{1}{�-�}��$

$=(1+�)\ln \mathrm{\mid }�-�\mathrm{\mid }+{�}_2$

Here, ${�}_2$ is another constant of integration.

Final Result:

Now, we can combine the results of both parts of the integral to obtain the final solution:

$\int \frac{�+1}{�-�}��=�+(1+�)\ln \mathrm{\mid }�-�\mathrm{\mid }+�$

Here, $�$ is the constant of integration, which combines the constants ${�}_1$ and ${�}_2$ from the previous integrations.

Conclusion:

In this exploration of the integral of $\frac{�+1}{�-�}$, we demonstrated the use of polynomial division as a powerful tool for simplifying complex rational functions. The resulting integral was broken down into two parts: the integral of a constant, which yielded a linear term, and the integral of a logarithmic function, which provided a logarithmic term.

The ability to manipulate and solve integrals involving rational functions is essential in various fields of mathematics, physics, and engineering. Such techniques form the foundation of more advanced calculus and mathematical analysis, allowing us to tackle a wide range of mathematical problems.

As you delve deeper into the world of calculus, you'll discover that the integration of rational functions is just one of the many fascinating challenges that await, showcasing the elegance and power of mathematical techniques.