Problem 14 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=\frac{2+x^{2}}{1+x^{2}}$$

Step-by-Step Solution

Answer

The most general antiderivative is \( x + \tan^{-1}(x) + C \).

Step 1: Recognize the Structure of the Integral

Examine the function \( f(x) = \frac{2 + x^2}{1 + x^2} \). Notice that it can be split into simpler fractions. By dividing the numerator: \( \frac{2 + x^2}{1 + x^2} = \frac{2}{1 + x^2} + \frac{x^2}{1 + x^2} \).

Step 2: Integrate Each Term Separately

Now, integrate each term separately. Consider \( \int \frac{2}{1 + x^2} \, dx \) and \( \int \frac{x^2}{1 + x^2} \, dx \). For the first term, \( \int \frac{2}{1 + x^2} \, dx = 2 \tan^{-1}(x) + C_1 \), using the standard integral formula for \( \frac{1}{1+x^2} \).For the second term, use the identity \( \frac{x^2}{1 + x^2} = 1 - \frac{1}{1 + x^2} \). Thus,\( \int \frac{x^2}{1 + x^2} \, dx = \int \left(1 - \frac{1}{1 + x^2}\right) \, dx \ = \int \, dx - \int \frac{1}{1 + x^2} \, dx = x - \tan^{-1}(x) + C_2 \).

Step 3: Combine the Integrals

Combine the results of the integrals from Step 2:\( 2 \tan^{-1}(x) + x - \tan^{-1}(x) + C_1 + C_2 = x + \tan^{-1}(x) + C \), where \( C = C_1 + C_2 \) is the constant of integration.

Step 4: Check by Differentiating the Antiderivative

Differentiate the antiderivative to verify:\( \frac{d}{dx} \left( x + \tan^{-1}(x) + C \right) = 1 + \frac{1}{1 + x^2} \).This simplifies to \( \frac{2 + x^2}{1 + x^2} = f(x) \), confirming that the antiderivative is correct.

Key Concepts

Integration Techniques

When faced with solving integrals, especially for complex functions, using appropriate integration techniques is crucial. One of the fundamental methods involves recognizing the structure of the function you want to integrate. For instance, in the original exercise, the function \[f(x) = \frac{2 + x^2}{1 + x^2}\] can be split into simpler, more manageable parts. By expressing it as two separate fractions—\[\frac{2}{1 + x^2} + \frac{x^2}{1 + x^2}\]—you can approach each part individually. This transformation simplifies the integration process, as each term is easier to handle on its own.

• Use partial fraction decomposition when possible to simplify complex fractions.
• Identify standard forms of integrals, such as \( \frac{1}{1+x^2} \), which corresponds to the arctan function.
• Always look for algebraic identities, like expressing \( \frac{x^2}{1 + x^2} \) as \( 1 - \frac{1}{1 + x^2} \), which can simplify the process further.

These techniques are not only helpful in tackling specific problems but also serve as a foundation for solving more complex integrals involving trigonometric, exponential, or logarithmic functions.

Differentiation

Differentiation is the process of finding the derivative of a function, which is fundamental in calculus. In the context of verifying an antiderivative, differentiation acts as a powerful tool. After determining the antiderivative, you differentiate it to ensure that it corresponds to the original function.For example, having found the antiderivative \[x + \tan^{-1}(x) + C\]for the function \[f(x) = \frac{2 + x^2}{1 + x^2},\]you verify by differentiating it back to \[1 + \frac{1}{1 + x^2}.\]This demonstrates:

• The derivative of \( \tan^{-1}(x) \) is \( \frac{1}{1 + x^2} \), showing why arctangent terms appear in integrals.
• Checking the derivative of each component ensures accuracy.

This method of verification reinforces your solution and confirms correctness. It's an essential step in any problem involving antiderivatives.

Trigonometric Integrals

Trigonometric integrals often present themselves when dealing with functions involving the reciprocal of quadratic expressions, such as \( \frac{1}{1+x^2} \). In this specific situation, understanding the integral of this form reveals connections to trigonometric functions. The \(\int \frac{1}{1+x^2} \, dx\)results in \tan^{-1}(x) + C,\ linking trigonometric concepts directly into the problem. Similarly, recognizing identities such as trigonometric substitutions can vastly simplify integration:

• Identities like \( \tan^{-1}(x) + C \) suggest a geometric interpretation, which might be useful for visualization.
• Converting expressions into forms that directly lead to trigonometric integrals can expedite solving the problem.

Incorporating these ideas underscores the relevance of trigonometry in integrals, thus expanding your problem-solving toolkit and showing how connected different areas of mathematics truly are.