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Magazine Chapter 7: Problem 24
Perform the indicated integrations. $$ \int \frac{x^{3}}{x^{4}+4} d x $$
Short Answer
Expert verified
The integral is \( \frac{1}{4} \ln |x^4+4| + C \).
Step by step solution
01
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.
02
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:
03
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.
04
Integrate
Now, integrate \( \int \frac{1}{4u} \, du \). The integral of \( \frac{1}{u} \) is \( \ln |u| \), so:\[ \int \frac{1}{4u} \, du = \frac{1}{4} \ln |u| + C \].
05
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 \( x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( u = x^4 + 4 \).
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 \( du \): \( dx = \frac{du}{4x^3} \)
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:
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 \( \int \frac{1}{4u} \, du \).
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:
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.
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 \( \int \frac{1}{u} \, du = \ln |u| + C \):
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 \( x \): \( \frac{1}{4} \ln |x^4 + 4| + C \).
