Problem 6 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{1}{4+(x-3)^{2}} d x $$

Step-by-Step Solution

Answer

\(\int \frac{1}{4+(x-3)^{2}} dx = \frac{1}{2} \arctan(\frac{x - 3}{2}) + C \)

Step 1: Identify the Formula Relevant for Problem

The indefinite integral resembles the following standard formula in the variable substitution form \( \int \frac{1}{a^{2}+x^{2}} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C \) where \(a\) and \(C\) are constant. In our integral, \(a^{2} = 4\), hence \(a = 2\) and \( x = (x - 3) \).

Step 2: Substitute and Apply the Identified Formula

Applying the identified formula with \(a = 2\) and \(x = (x - 3)\), the indefinite integral simplifies as:\[\int \frac{1}{4+(x-3)^{2}} dx = \frac{1}{2} \arctan(\frac{x - 3}{2}) + C \]

Step 3: State the Final Answer

The final answer includes the constant of integration \(C\) because this is an indefinite integral. So the integral\[\int \frac{1}{4+(x-3)^{2}} dx = \frac{1}{2} \arctan(\frac{x - 3}{2}) + C \] is the solution.

Key Concepts

Arctangent Function

The arctangent function, often denoted as \(\arctan(x)\), is the inverse of the tangent function. It is used to determine the angle whose tangent is the given number. This function is vital in many integral problems where the result involves a trigonometric substitution.
The range of the arctangent function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), which means it outputs angles within this domain. In the context of integration, particularly with expressions such as \(\int \frac{1}{a^2 + x^2} dx\), the result can often be expressed in terms of the arctangent.
This specific integration formula links directly to this function because the derivative of \(\arctan(x)\) yields the form \(\frac{1}{1+x^2}\). Hence, understanding the arctangent function's role is crucial, especially in reverse-engineering derivatives back to integrals.

Definite vs Indefinite Integrals

In calculus, integration can be classified into two main types: definite and indefinite integrals. An indefinite integral does not involve any boundaries or limits, and it represents a family of functions. Its result is always expressed with an added constant \(C\), to encompass any vertical shift of the antiderivative. This is evident in the original problem, as the solution has \(+C\) added to it.

• Indefinite Integral: Represents the antiderivative of a function. It includes a constant of integration \(C\).
• Definite Integral: Calculated between two boundary values. It results in a specific numerical value.

Understanding the difference is essential, as definite integrals calculate areas under curves, while indefinite integrals focus on finding the general form of a function.
For the given exercise, we performed an indefinite integral, so the result includes \(C\), indicating that a variety of antiderivatives satisfy the integral.

Integration Formulas

Integration formulas are tools that simplify finding antiderivatives of functions. They often help transform complex integrals into easily solvable forms. Several standard integration formulas are derived based on derivatives of common functions.

• The integral \(\int \frac{1}{a^2+x^2} dx\) results in \(\frac{1}{a} \arctan{\frac{x}{a}} + C\). In our problem, recognizing this formula allowed us to quickly adapt to \(\int \frac{1}{4+(x-3)^2} dx\).
• The substitution technique is also vital, where a complex variable can be simplified through rephrasing in terms of another variable or number.
• Constant multiple rule: \(\int k\cdot f(x)\, dx = k \cdot \int f(x)\, dx\). This rule helps to simplify the integration of constants.

The use of these formulas, along with proper identification of the expression structure, enables efficient solving of integrals, turning potentially cumbersome calculations into straightforward operations. Mastery of these formulas is key for tackling calculus problems efficiently.