Question

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 d x}{x^{2}+1}$$

Short answer

The evaluated integral of \( \frac{2}{x^{2}+1} \) is \( 2 \cdot (\tan^{-1}x) + C \).

Step-by-step solution

Identify the Function

Recognize the integral as a function. The integrand is a quotient of a constant and a binomial, which is a rational function in \( x \).

Standard Form

The form of the integrand is the derivative of \( \tan^{-1}x \), which is \( \frac{1}{x^{2}+1} \). Therefore, we can rewrite the function as \( 2 \cdot \frac{1}{x^{2}+1} \). This form allows us to identify that the result of the integral will involve \( \tan^{-1}x \).

Evaluate the Integral

Use the table of standard integrals, which states that the integral of \( \frac{1}{x^{2}+1} \) is \( \tan^{-1}x + C \), where \( C \) is the constant of integration. Since our function is \( 2 \cdot \frac{1}{x^{2}+1} \), the result will be \( 2 \cdot (\tan^{-1}x) + C \).

Key concepts

Integrals of Rational Functions

Understanding how to evaluate integrals of rational functions is crucial when working with expressions that involve ratios of polynomials. A rational function is any function that can be represented as the ratio of two polynomial functions. In the case of \( \int \frac{2 dx}{x^2+1} \), the integrand is a simple ratio where the numerator is a constant, 2, and the denominator is a polynom \(x^2+1\), which is a sum of squares pattern.

When integrating rational functions, it’s important to recognize forms that match the derivatives of known functions. This both simplifies the problem and directly leads us to a solution using standard results or a table of integrals. In our problem, the integral has the same form as the derivative of the inverse tangent function, which is a signal that inverse trigonometric identities might be involved in the solution.

Table of Standard Integrals

A table of standard integrals is an invaluable tool for solving integration problems. It lists the integrals of common functions, helping you to quickly identify known integral forms and their results without having to perform the integration process from scratch every time. In the given exercise, recognizing that \( \frac{1}{x^2+1} \) matches an entry in the standard integrals table speeds up the evaluation process immensely.

To optimize learning and recalling, it's beneficial for a student to get familiar with these standard results. Some of the most frequently used integrals from this reference include those for basic power functions, exponential functions, and trigonometric functions. Matching integrands to this list is a key skill for swiftly solving integrals.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \(\tan^{-1}x\), also known as arctan(x), allow us to integrate certain rational functions easily. These functions are the inverses of trigonometric functions and are used when we need to find the angle whose trigonometric function equals a given number.

For instance, the derivative of \(\tan^{-1}x\) is \(\frac{1}{x^2+1}\). Hence, when faced with the integral of this derivative, we can imply that \(\int \frac{1}{x^2+1} dx = \tan^{-1}x + C\). In our integral, we exploit this knowledge to conclude that \(\int \frac{2 dx}{x^2+1} = 2 \tan^{-1}x + C\), where \(C\) represents the constant of integration. Understanding these relationships expands a student's capability to tackle more complex integration problems.

Constant of Integration

The constant of integration, denoted as \(C\), plays a fundamental role in indefinite integrals. This is because the process of integration is essentially the reverse of differentiation, and when we differentiate a constant, it becomes zero. Thus, when we integrate, we must account for any possible constant that was lost in the differentiation process.

In practice, whenever we integrate a function and find its antiderivative, we add \(C\) to signify that there could be an infinite number of antiderivatives, each corresponding to a different constant value. The correct value of \(C\) is typically determined by initial conditions or boundary values provided in a specific problem context, which can transform the indefinite integral into a definite one with exact bounds.