Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{x^{3}}{\sqrt{1-x^{4}}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(-e^{2/x} + C\).

Step 1: Perform Substitution

Introduce a substitution for the integral. Let \(u = 2 / x\). Then find \(du\) in terms of \(dx\). The derivative of \(u\) with respect to \(x\) is \(-2 / x^{2}\). So, \(du = -2 / x^{2} dx\). Therefore, the given integral can be re-written as \(-\int e^u du\).

Step 2: Solve Integral

The integral \(-\int e^u du\) is a standard integral which can be integrated directly. The integral of \(e^u\) with respect to \(u\) is \(e^u\). Therefore, the integral becomes \(-e^u + C\), where \(C\) is the constant of integration.

Step 3: Substitute Back

Now substitute back \(u = 2 / x\) to convert the solution into the original variable, \(x\). Therefore, our final solution will be \(-e^{2/x} + C\).

Key Concepts

Symbolic Integration

Symbolic integration is the process of finding the antiderivative or indefinite integral of mathematical functions in terms of symbols, rather than numbers.

Unlike numerical integration, where we calculate the integral's value at certain points, symbolic integration involves a general solution that applies to all possible values within the domain of the function.Within the context of our exercise, symbolic integration allows us to find a function whose derivative is \( \frac{1}{x^{2}} e^{\frac{2}{x}} \) without specifying the limits of integration. This resulting function is represented as an expression that includes an integration constant, \(C\), to account for the family of all possible antiderivatives.

U-Substitution

U-substitution is a technique used in integration, which simplifies complex integrals by substituting a part of the integral with a new variable, \( u \). This method often transforms a difficult integral into a simpler one that we can solve more readily.

In the example given, \( u = \frac{2}{x} \) is chosen for substitution to facilitate the integration of \( \frac{1}{x^2} e^{\frac{2}{x}} \). The differential \( du \) is calculated by differentiating \( u \) with respect to \( x \), yielding \( du = -\frac{2}{x^2} dx \).

The integral is then expressed in terms of the new variable \( u \), resulting in a simpler integral of \( -\int e^u du \), which can be directly integrated. U-substitution not only helps solve the integral in question but is also essential in reversing the process. After finding the antiderivative in terms of \( u \), we need to 'substitute back' to return to the original variable, \( x \), as shown in the final step of the problem.

Integration Constants

In the world of calculus, integration constants play a critical role in indefinite integrals. Whenever an antiderivative is determined, it is accompanied by a constant, typically denoted as \(C\).

The reason for this constant is simple yet profound: an indefinite integral represents a family of functions, all of which possess the same derivative but differ by a constant term. For instance, the functions \(F(x) = x^2 + 1\) and \(G(x) = x^2 + 5\) have the same derivative of \(2x\), yet they are clearly distinct functions because they are offset by a constant. The integration constant \(C\) accounts for this range of possibilities.

Therefore, when we solve an indefinite integral such as \(-\int e^u du\), the result \(-e^u + C\) indicates that there is not one, but an infinite number of antiderivatives, each differing by a constant value. Hence, the integration constant is essential in conveying the complete set of solutions for an indefinite integral.