(x^3-1)/(x-1) Integral

3 min read Jun 17, 2024

Evaluating the Integral of (x³ - 1)/(x - 1)

This article will guide you through the process of finding the integral of the function (x³ - 1)/(x - 1). We will utilize various techniques, including polynomial long division and basic integration rules.

Understanding the Function

The expression (x³ - 1)/(x - 1) represents a rational function, a function where the numerator and denominator are both polynomials. It is crucial to simplify this function before attempting integration.

Simplifying Using Polynomial Long Division

To simplify the expression, we can perform polynomial long division. Here's how:

Set up the division:

     x² + x + 1
   x - 1 | x³ + 0x² + 0x - 1
         -(x³ - x²)
         ----------------
               x² + 0x
               -(x² - x)
               -------------
                       x - 1
                       -(x - 1)
                       ---------
                           0

Result: The division yields a quotient of x² + x + 1 and a remainder of 0. This means: (x³ - 1)/(x - 1) = x² + x + 1

Integrating the Simplified Expression

Now, we can easily integrate the simplified expression:

∫(x³ - 1)/(x - 1) dx = ∫(x² + x + 1) dx

Applying the power rule of integration:

∫(x² + x + 1) dx = (x³/3) + (x²/2) + x + C

Therefore, the integral of (x³ - 1)/(x - 1) is (x³/3) + (x²/2) + x + C, where C is the constant of integration.

Key Takeaways

Simplifying rational functions through polynomial long division is essential for easier integration.
The power rule of integration is crucial for integrating polynomial terms.
Remember to always add the constant of integration, C, when evaluating indefinite integrals.

This example showcases how basic algebraic manipulations combined with standard integration techniques can effectively solve seemingly complex integrals.