Integrating using long division and completing the square

Certainly! Here’s a brief summary of the two concepts:

Integrating Using Long Division:

When integrating rational functions, if the degree of the numerator is greater than or equal to the degree of the denominator, you can use polynomial long division to simplify the function. The idea is to divide the numerator by the denominator, resulting in a polynomial (the quotient) and a remainder. You can then express the integral as the sum of the integral of the quotient (which can be integrated directly) and the integral of the remainder divided by the original denominator.

Completing the Square:

This technique is often used to simplify quadratic expressions in integrals. It involves rewriting a quadratic expression in the form $ax^2 + bx + c$ as $a(x - h)^2 + k$ , where $h$ and $k$ are constants determined from the original coefficients. Completing the square makes it easier to integrate expressions that involve square roots or can be transformed into a standard form, such as those resembling the integral of $\frac{1}{x^2 + a^2}$ or $e^{-(x-h)^2}$ .

Both methods are useful for transforming complicated integrals into more manageable forms.

In this article

Part 1: Integration using long division Part 2: Integration using completing the square and the derivative of arctan(x)

Part 1: Integration using long division

Here we do polynomial long division to make an integral more computable.

When studying "Integration using long division," focus on the following key points:

Understanding the Process:
- Recognize that long division is used when the degree of the numerator is greater than or equal to the degree of the denominator in a rational function.
Performing Long Division:
- Divide the polynomial in the numerator by the polynomial in the denominator.
- Write down the quotient and the remainder.
Rewriting the Function:
- Express the original rational function as a sum of the quotient and the remainder divided by the original denominator.
Integrating the Result:
- Integrate the quotient directly.
- Integrate the remainder divided by the denominator using standard integration techniques.
Common Techniques:
- Familiarize yourself with techniques for integrating simpler fractions that result from the long division process, such as polynomial, logarithmic, or trigonometric integrals.
Practice Problems:
- Work through various examples to solidify your understanding and gain proficiency in the long division process and subsequent integration.

By mastering these points, you'll be able to effectively apply long division to integrate rational functions.

Part 2: Integration using completing the square and the derivative of arctan(x)

Sometimes we can integrate rational functions by using the method of completing the square in the denominator and then integrating using u-substitution and our knowledge about the derivative of arctan(x).

Certainly! Here are the key points to focus on:

Completing the Square

Purpose: Used to transform quadratic expressions into a perfect square form to simplify integration.
Formula: For $ax^2 + bx + c$ , we can rewrite it as $a \left( x + \frac{b}{2a} \right)^2 + k$ , where $k$ adjusts the constant term.
Steps:
- Factor out the coefficient of $x^2$ if $a \neq 1$ .
- Find the value to complete the square: $\left( \frac{b}{2a} \right)^2$ .
- Adjust the constant term accordingly.
Applications in Integration: Facilitates integrals of the form $\int \frac{1}{ax^2 + bx + c}$ by converting to a format amenable for trigonometric substitution or basic integration techniques.

Derivative of $\arctan(x)$

Formula: The derivative is given by $\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}$ .
Importance in Integration: The derivative relates to integrals such as $\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$ , providing a method for solving certain types of integrals.
Graphical Interpretation: Represents the slope of the tangent to the curve $y = \arctan(x)$ ; it approaches $\frac{\pi}{2}$ and $-\frac{\pi}{2}$ as $x$ tends to $\infty$ and $-\infty$ , respectively.

Integration Techniques

Using Completing the Square: After rewriting a quadratic, integrals can often be solved through substitution or varying methods based on the transformed expression.
Connection to Arctangent: Recognizing forms which relate to the derivative of $\arctan(x)$ can simplify integration processes, especially when dealing with rational functions.

Summary

Master the technique of completing the square to manipulate quadratic expressions.
Understand the derivative of $\arctan(x)$ and its integration implications.
Apply these concepts systematically in integrals involving rational functions and quadratic forms.

These points should provide a focused framework for studying the integration techniques involving completing the square and the application of the derivative of arctangent.

Integrating using long division and completing the square