Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{x^{3}}{x^{4}+4} d x\)

Step-by-Step Solution

Answer

\( \int \frac{x^3}{x^4 + 4} \, dx = \frac{1}{4} \ln |x^4 + 4| + C \).

Step 1: Identify the Integration Technique

This integral involves a rational function where the degree of the numerator is less than the degree of the denominator. The best approach is to use substitution methods to simplify the integration.

Step 2: Choose a Substitution

Choose the substitution that will simplify the denominator. Let \( u = x^4 + 4 \). Then, the differential \( du = 4x^3 \, dx \). This implies that \( x^3 \, dx = \frac{1}{4} du \).

Step 3: Substitute and Simplify the Integral

Replace \( x^3 \, dx \) with \( \frac{1}{4} du \) and \( x^4 + 4 \) with \( u \). The integral becomes \[ \int \frac{1}{4} \frac{1}{u} \, du \]. Simplifying, it is \( \frac{1}{4} \int \frac{1}{u} \, du \).

Step 4: Integrate

The integral \( \int \frac{1}{u} \, du \) is the natural logarithm: \( \ln |u| + C \). Therefore, \[ \frac{1}{4} \ln |u| + C \].

Step 5: Substitute Back

Substitute back \( u = x^4 + 4 \) into the integrated expression to get the final answer: \[ \frac{1}{4} \ln |x^4 + 4| + C \].

Key Concepts

Substitution Method

The substitution method is a powerful technique for simplifying the integration process. It works by transforming a complex integral into a more manageable form. You choose a substitution that transforms the original variable, making the integral easier to evaluate.

When performing substitution, follow these steps:

• Identify a part of the integrand that can be replaced by a single variable. This often involves a composite function or nested terms.


• Choose a substitution, usually a portion of the integrand, such as the denominator or an awkward exponent. For this exercise, we chose: \( u = x^4 + 4 \).


• Compute the differential, \( du \), of your chosen substitution. Differentiate the substitution function with respect to the original variable. In this example: \( du = 4x^3 \, dx \).


• Express \( dx \) or any other variable term back in terms of \( du \) to simplify the integral.

The goal is to convert the original integral into one involving the new variable \( u \), solve it, then revert back to the original variable for the final result.

Rational Functions

Rational functions are expressions that have polynomials in both their numerator and denominator. They take the form \( P(x)/Q(x) \), where \( P(x) \) and \( Q(x) \) are polynomials. Understanding their behavior is crucial in integration because simpler rational functions can often be tackled with straightforward methods.

Key points about rational functions:

• When the degree of the polynomial in the numerator is less than the degree in the denominator, as is the case in this exercise, substitution or partial fraction decomposition might simplify the integration.


• Complex fractions and higher degree polynomials may require algebraic manipulation first; sometimes, polynomial long division is needed before applying integration techniques.

In our integral \( \int \frac{x^3}{x^4+4} \, dx \), the degree condition between numerator and denominator is perfect for substitution. The substitution transformed the rational function into an easier form that could be solved using basic logarithmic integration.

Definite and Indefinite Integrals

Integration is divided into definite and indefinite categories, each with distinct outcomes and uses. In this exercise, we were dealing with an indefinite integral.

Here's what distinguishes them:

• Indefinite Integrals: Represent a family of functions and include a constant of integration, denoted often as \( C \). This constant accounts for all possible vertical shifts of the antiderivative. For the integral \( \int \frac{x^3}{x^4+4} \, dx \), the solution will include \( +C \) after finding the antiderivative.


• Definite Integrals: Evaluate the area under the curve over a specific interval, resulting in a real number without an arbitrary constant. Definite integrals are computed using the fundamental theorem of calculus and require upper and lower limits of integration.

Both types are vital in calculus, providing tools for solving problems across physics, engineering, and beyond.