Problem 44 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \frac{x}{1 + x^4} \, dx \)

Step-by-Step Solution

Answer

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

Step 1: Identify Substitution

To evaluate the indefinite integral, notice that the expression in the denominator, \(1+x^4\), suggests evaluating whether a substitution can simplify it. Consider the substitution \( u = x^2 \). This implies \( du = 2x \, dx \), or equivalently \( x \, dx = \frac{1}{2} du \).

Step 2: Express Original Integral with Substitution

Using the substitution \( u = x^2 \), rewrite the integral. The integral becomes: \[ \int \frac{x}{1+x^4} \, dx = \int \frac{x}{1+(x^2)^2} \, dx \] Substitute \( u = x^2 \) and \( x \, dx = \frac{1}{2} du \) to get: \[ \int \frac{1}{2} \frac{1}{1+u^2} \, du \].

Step 3: Recognize Standard Integral

The integral \( \int \frac{1}{1+u^2} \, du \) is a standard form which results in \( \tan^{-1}(u) \). Use this knowledge to evaluate the integral: \[ \frac{1}{2} \int \frac{1}{1+u^2} \, du = \frac{1}{2} \tan^{-1}(u) + C \], where \(C\) is the integration constant.

Step 4: Substitute Back to Original Variable

Replace \( u \) back with \( x^2 \) to return to the original variable. Thus, \( \frac{1}{2} \tan^{-1}(u) + C \) becomes \( \frac{1}{2} \tan^{-1}(x^2) + C \).

Step 5: Write Down the Final Answer

The evaluated indefinite integral is: \[ \int \frac{x}{1+x^4} \, dx = \frac{1}{2} \tan^{-1}(x^2) + C \]. This is the solution to the integral in terms of the original variable, \(x\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in integration, often used to simplify complex integrals by changing variables. In this problem, we chose the substitution \( u = x^2 \). This decision helps transform the integral into a simpler form. When you choose a substitution, it's important to also express the differential in the new variable. Here, since \( du = 2x \, dx \), it follows that \( x \, dx = \frac{1}{2} du \). This ability to rewrite both the integral and its components in terms of \( u \) streamlines the process.Using substitution:

• Simplifies the integral's appearance.
• Makes complex expressions more manageable.
• Allows the use of known basic integrals and their solutions.

Always remember to convert the integral back to the original variable at the end. This ensures the solution is in terms of the original problem statement.

Integration Techniques

Integration techniques encompass various strategies used to solve integrals. Substitution, as we used here, is one of them. Other popular techniques include parts, partial fractions, and trigonometric identities. Choosing the correct technique:

• Depends on the form of the integrand.
• Requires pattern recognition and algebraic manipulation.

In this particular exercise, substitution simplified the integral into a form that resembles a basic integral we are familiar with. Recognizing when to apply a technique is always influenced by practice and familiarity with different integral forms.

Inverse Trigonometric Function

Inverse trigonometric functions, like arctan or \( \tan^{-1} \), often appear when dealing with integrals involving functions of the form \( \frac{1}{1+u^2} \). In our solution, integrating \( \frac{1}{1+u^2} \) results in \( \tan^{-1}(u) \).Key aspects:

• These functions return angles whose trigonometric functions result in the original input.
• Recognize patterns in integrals that match the derivatives of inverse trig functions to simplify integration.

Using this knowledge allows us to determine the result of an integral without deep computation. With practice, identifying these inverse trigonometric forms becomes intuitive.

Integration Constant

In indefinite integrals, the integration constant \( C \) is crucial. It represents an arbitrary constant because indefinite integration yields a family of functions. When you integrate, especially in indefinite terms, it’s important always to include \( C \) in your answer. This constant accounts for all vertical shifts of the antiderivative.Why \( C \) matters:

• Ensures that all potential solutions to a differential equation are represented.
• Reflects the indefinite nature of the integration process.

Skipping this step can lead to incorrect understanding or application of the integral’s solution, particularly in physics and engineering problems where precise values are determined later on from boundary conditions.