Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{x^{3}}{x^{4}+4} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{4} \ln |x^4+4| + C \).

Step 1: Identify the Integration Technique

To solve the integral \( \int \frac{x^3}{x^4 + 4} \, dx \), notice that the degree of the polynomial in the numerator is one less than the degree of the polynomial in the denominator. This suggests the use of a substitution method. A suitable substitution will simplify the integrand.

Step 2: Apply Substitution

Let \( u = x^4 + 4 \). Then, differentiate both sides with respect to \( x \) to find \( du \): \( du = 4x^3 \, dx \). Solving for \( dx \) gives \( dx = \frac{du}{4x^3} \). Substitute these into the integral:

Step 3: Substitute and Simplify

Substitute \( u = x^4 + 4 \) and \( dx = \frac{du}{4x^3} \) into the integral:\[ \int \frac{x^3}{u} \cdot \frac{du}{4x^3} = \int \frac{1}{4u} \, du \]. The \( x^3 \) terms cancel out, simplifying the integral greatly.

Step 4: Integrate

Now, integrate \( \int \frac{1}{4u} \, du \). The integral of \( \frac{1}{u} \) is \( \ln |u| \), so:\[ \int \frac{1}{4u} \, du = \frac{1}{4} \ln |u| + C \].

Step 5: Back-Substitute

Replace \( u \) with the substitution back in terms of \( x \):\[ \frac{1}{4} \ln |x^4+4| + C \]. This provides the original integral in terms of the variable \( x \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in integration used to simplify complex integrals by changing variables. When faced with an integral like \( \int \frac{x^{3}}{x^{4}+4} \, dx \), the strategy involves picking a substitution that makes the integral more manageable.
For this specific exercise, we set \( u = x^4 + 4 \).

• This substitution simplifies the variable relationships. Here, the derivative \( du = 4x^3 \, dx \) helps us replace \( dx \) in the integral.
• Next, express \( dx \) in terms of \( du \): \( dx = \frac{du}{4x^3} \)

The goal is to reformulate the original integral, reducing it to a simpler form. This often results in a basic integral that is easier to solve.

Polynomial Integration

Polynomial integration involves integrating expressions where the variables are raised to whole number powers.
In this case, although we started with a polynomial division \( \int \frac{x^3}{x^4 + 4} \, dx \), it was advantageous to apply substitution.
This created a simplified scenario that transforms polynomial components:

• Using \( u = x^4 + 4 \), the complexity of the polynomial in the denominator was hidden behind the new variable \( u \).
• The original polynomial integral translates into \( \int \frac{1}{4u} \, du \).

Thus, polynomial integration sometimes demands symbol manipulation like substitution to see through intricate variable interactions.

Integral Simplification

Integral simplification is about reducing an integral to its most straightforward form, allowing easier calculation.
In the exercise \( \int \frac{x^3}{x^4 + 4} \, dx \), our substitution \( u = x^4 + 4 \) turned a complex integrand into a manageable one:

• The transition via substitution turned the integral into \( \int \frac{1}{4u} \, du \).
• By canceling terms \( x^3 \, \text{and} \, x^3 \) from numerator and denominator, we arrived at this reduction.

Such simplification is crucial for tackling integrals that are otherwise intractable through direct methods.

Logarithmic Integration

Logarithmic integration occurs when the integrand can be transformed into a function whose integral involves a natural logarithm.
After substitution, the integral \( \int \frac{1}{4u} \, du \) was simplified. This integral matches the form \( \int \frac{1}{u} \, du = \ln |u| + C \):

• The factor \( \frac{1}{4} \) is a constant that can be pulled out, leaving us with \( \frac{1}{4} \ln |u| + C \) as the result.
• We then reverse our substitution to express \( u \) back in terms of \( x \): \( \frac{1}{4} \ln |x^4 + 4| + C \).

Logarithmic integration is commonly used when dealing with expressions that simplify into a form suitable for natural logarithm application.