Problem 32 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \frac{x}{x^2 + 4} \, dx \)

Step-by-Step Solution

Answer

\( \frac{1}{2} \ln |x^2 + 4| + C \)

Step 1: Identify the integration method

The integral \( \int \frac{x}{x^2 + 4} \, dx \) involves a rational function which suggests the use of "Substitution Method" since the derivative of the denominator \( x^2 + 4 \) is related to the numerator \( x \).

Step 2: Define the substitution

Let \( u = x^2 + 4 \). Then, compute the derivative \( \frac{du}{dx} = 2x \) which implies \( du = 2x \, dx \). Solve for \( x \, dx \) to get \( x \, dx = \frac{1}{2} \, du \).

Step 3: Rewrite the integral with the substitution

Substitute \( u \) and \( du \) into the integral: \[ \int \frac{x}{x^2 + 4} \, dx = \int \frac{1}{2} \cdot \frac{1}{u} \, du \]. This simplifies to \( \frac{1}{2} \int \frac{1}{u} \, du \).

Step 4: Integrate with respect to u

Integrate \( \frac{1}{u} \, du \) to get \( \ln |u| + C \), where \( C \) is the constant of integration. Thus the expression becomes \( \frac{1}{2} (\ln |u| + C) \).

Step 5: Back-substitute for x

Replace \( u \) with \( x^2 + 4 \) to revert the substitution. This gives \( \frac{1}{2} \ln |x^2 + 4| + C \).

Step 6: Final Answer

The indefinite integral \( \int \frac{x}{x^2 + 4} \, dx \) equals \( \frac{1}{2} \ln |x^2 + 4| + C \).

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. When dealing with indefinite integrals, we seek to find the antiderivative or the original function of the derivative provided. Various techniques are available depending on the form of the function, such as:

• Basic antiderivatives
• Substitution method
• Integration by parts
• Partial fraction decomposition

Choosing the right technique is crucial for solving integrals efficiently. In this exercise, we identified the integral of a rational function which often suggests the use of substitution method.

Substitution Method

The substitution method is a powerful technique for simplifying integrals, particularly when the integral involves a composite function.
In our exercise, we have the integral \( \int \frac{x}{x^2 + 4} \, dx \). We notice that the derivative of the denominator \( x^2 + 4 \) is related to the numerator \( x \). So, we can set a substitution: let \( u = x^2 + 4 \). Then the derivative \( \frac{du}{dx} = 2x \) lets us express \( du = 2x \, dx \). Solving for \( x \, dx \) gives \( x \, dx = \frac{1}{2} \, du \).

This transforms the integral into a simpler form, allowing us to integrate in terms of \( u \). The substitution method effectively reduces complexity by changing variables, making integration more manageable.

Rational Functions

Rational functions are quotients of two polynomials. In our exercise, the rational function \( \frac{x}{x^2 + 4} \) is present. Working with rational functions often requires certain techniques of integration like substitution or, sometimes, partial fractions for more complex denominators.

The polynomial structure of the numerator \( x \) and the denominator \( x^2 + 4 \), and the relationship between their derivatives, guided us toward using substitution. By recognizing rational functions, we determine their integration paths more intuitively, often simplifying the problem dramatically.

Derivative

Understanding derivatives is crucial when working with integrals, such as when employing the substitution method. Derivatives provide us with the rate of change of a function, and when we solve integrals, we seek an antiderivative, which essentially reverses differentiation.

In this context, recognizing that the derivative of \( x^2 + 4 \) is \( 2x \) directly informs the substitution method we choose. It enables us to replace parts of our integral to simplify the process. Mastery of derivatives is fundamental in integrating functions, as it bridges the transition between differentiation and integration effectively.

Constant of Integration

When dealing with indefinite integrals, as in our exercise, the constant of integration \( C \) plays a vital role. Indefinite integrals do not specify limits, thus the result represents a family of functions differing by a constant.

Adding \( C \) accounts for any constant that might have been in the original function before differentiation. In our solution, after integrating \( \int \frac{1}{u} \, du \), we obtain \( \ln |u| + C \) representing all possibilities of the antiderivative. This highlights the inherent uncertainty in reversing differentiation without boundaries.