Evaluating Integrals Involving Polynomial Denominators: A Comprehensive Guide

When faced with integrals involving polynomial denominators, the use of integration by substitution can simplify the process. This guide will walk you through the steps for evaluating the integral (int frac{x}{1 x^4} , dx). We will explore multiple methods to solve this integral and highlight the significance of the substitution technique.

Introduction to the Integral

Consider the integral (int frac{x}{1 x^4} , dx). At first glance, this integral may appear complex, especially given the quartic term in the denominator. However, with the appropriate substitution, we can transform this integral into a more manageable form.

Method 1: Substitution Using (u x^2)

In this method, let's start by setting (u x^2). Differentiating both sides, we get:

[du 2x , dx quad implies quad dx frac{du}{2x}]

Substituting (u x^2) and (dx frac{du}{2x}) into the integral, we obtain:

[int frac{x}{1 u^2} cdot frac{1}{2x} , du frac{1}{2} int frac{1}{1 u^2} , du]

We know that the integral of (frac{1}{1 u^2}) is (arctan(u)). Therefore:

[frac{1}{2} int frac{1}{1 u^2} , du frac{1}{2} arctan(u) C]

Substituting back (u x^2), we get:

[frac{1}{2} arctan(x^2) C]

Method 2: Substitution Using (u 2x^2)

Another method is to set (u 2x^2). Differentiating, we have:

[du 4x , dx quad implies quad dx frac{du}{4x}]

However, this method is less direct, and it may complicate the integral further. Instead, we will use the substitution (u x^2) as it is more straightforward.

Method 3: Direct Substitution Using (u x^2)

A simpler and more efficient method is to directly substitute (u x^2) and (du 2x , dx). This leads to:

[frac{1}{2} int frac{1}{1 u^2} , du frac{1}{2} arctan(u) C]

Substituting back (u x^2), we get:

[frac{1}{2} arctan(x^2) C]

Conclusion

In conclusion, the integral (int frac{x}{1 x^4} , dx) can be effectively evaluated using the substitution technique. By setting (u x^2), we transformed the integrand into a form that can be readily integrated. The final result is:

[boxed{frac{1}{2} arctan(x^2) C}]

Understanding and mastering the art of integration by substitution can significantly simplify the process of solving complex integrals. This method is widely applicable and forms the foundation for solving more advanced integrals in calculus.