Problem 57 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{d x}{2+(x-1)^{2}}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{x-1}{\sqrt{2}} \right) + C \).

Step 1: Identify the Integral Form

The integral is given as \( \int \frac{dx}{2 + (x - 1)^2} \). This integral is in the form that suggests the use of a trigonometric substitution or direct recognition as an inverse trigonometric function.

Step 2: Recognize the Inverse Trigonometric Function

The integral \( \int \frac{dx}{a^2 + x^2} \) is a standard form that results in \( \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \). Here, compare \( 2 + (x-1)^2 \) to \( a^2 + (x)^2 \). Observe that \( a^2 = 2 \) and note the shift \( (x-1) \), which suggests a substitution.

Step 3: Make the Substitution

Let \( u = x - 1 \), then \( du = dx \). The integral becomes \( \int \frac{du}{2 + u^2} \). This looks more like the standard inverse trigonometric form.

Step 4: Apply the Inverse Trigonometric Integration Formula

Using the formula \( \int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \left( \frac{u}{a} \right) + C \), identify \( a^2 = 2 \), so \( a = \sqrt{2} \). The integral becomes \( \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{u}{\sqrt{2}} \right) + C \).

Step 5: Substitute Back to Original Variable

Since \( u = x - 1 \), substitute back to get \( \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{x-1}{\sqrt{2}} \right) + C \).

Key Concepts

Inverse Trigonometric Functions

When dealing with integrals, recognizing standard forms is crucial, and inverse trigonometric functions are among these key forms. The inverse trigonometric functions, such as \( \tan^{-1}(x) \), \( \sin^{-1}(x) \), and \( \cos^{-1}(x) \), are often used to solve integrals involving quotients that resemble these functions' derivatives.
In our exercise, the integral \( \int \frac{dx}{2 + (x - 1)^2} \) suggests a relationship with the arctangent function, as it transforms into the form \( \int \frac{dx}{a^2 + x^2} \).

• In the formula \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) + C \), \( a \) represents a constant that, once identified, allows for direct application of the arctangent integration result.
• This form is particularly common and should be committed to memory, as it simplifies and expedites integral evaluations involving quadratic denominators.

Recognizing these patterns helps streamline the integration process, avoiding unnecessary complexity.

Integration Techniques

Successful integration often depends on spotting which technique will simplify the problem, whether it's recognizing standard forms or using a substitution method. Here, we leverage recognition of a standard inverse trigonometric form to navigate the integration process efficiently.
We begin by identifying that our integral resembles an arctangent form \( \int \frac{dx}{a^2 + x^2} \), indicating a connection to \( \tan^{-1}(x) \).
Key steps for using this technique effectively include:

• Identify the constants matching the standard form in the integrand, such as \( a^2 \) or shifts like \( (x-1) \).
• Consider expanding or rearranging the given integral to match a recognizable pattern before attempting substitutions.

Recognizing these techniques can drastically reduce both the complexity and time required for solving integral problems.

Substitution Method

The substitution method serves as a powerful tool in integration, especially when faced with integrals that deviate from pure standard forms. This technique involves replacing part of the integral with a single variable, simplifying the expression.
In our solution, we made the substitution \( u = x - 1 \), which simplified \( 2 + (x - 1)^2 \) to \( 2 + u^2 \). This change made the integral resemble a basic arctangent form. The key steps in this process are:

• Identify substitutions that simplify the function or align it more closely with a standard form.
• Remember to adjust the differential \( dx \) accordingly to ensure proper substitution throughout the integral.
• After integrating, replace the substitution variable back to reflect the original variable terms.

This method often bridges the gap between complex integrals and recognizable standard forms, making them much more approachable to integrate.