Integral of 3x^2 - 2x - 1 / x^3 - x: A Step-by-Step Guide

The integral of a rational function involving polynomials is a fundamental concept in calculus. This article provides a detailed, step-by-step guide to solving an integral expression involving such a function: finding the integral of 3x^2?2x?1/x^3?x^1.

Simplifying the Expression

First, let's start with the given expression:

3x^2?2x?1x^3?x

We can simplify the numerator and the denominator:

3x^2?2x?1x^3?x3x^2?3x 1x?1x *(x^2?1)

Partial Fraction Decomposition

We can decompose this into partial fractions: 3x^2?2x?1x^3?xAx Bx^2 Cx^2

The goal now is to find the constants (A) and (B).

Integration Steps

We integrate the individual terms:

∫Axdx ∫Bx^2dx ∫Cx^2dxlnx B Cx?1 clnx?B c

Solution Summary

The integral of 3x^2?2x?1/x^3?x can be written as:

lnx?2ln(x?1) c

This integral involves the natural logarithm function and is often used to further explore properties of logarithms and polynomials in calculus.

Conclusion

This step-by-step guide provided a thorough and structured approach to solving the integral of the given rational expression. Understanding such techniques is crucial for both educational and practical applications in the field of mathematics and engineering.