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Chapter 8: Problem 30

$$\text {Evaluate the following integrals.}$$ $$\int \frac{21 x^{2}}{x^{3}-x^{2}-12 x} d x$$

Short Answer

Expert verified
Question: Evaluate the integral 鈭�(21x^2) / (x^3 - x^2 -12x) dx Answer: The integral evaluates to 21x + C.

Step by step solution

01

Perform long division if necessary

Since the degree of the numerator is equal to the degree of the denominator, we must perform polynomial long division. However, upon closer examination, it's clear that the numerator is divisible by the denominator without any remainder. This simplifies the work as shown: $$\frac{21 x^2}{x^3 - x^2 - 12x} = \frac{21x^2}{x(x^2-x-12)} = 21$$ Now our integral becomes: $$\int 21 dx$$
02

Integrate term by term

The integral is now simple to calculate since it's just a constant. Integrating with respect to x, we get: $$\int 21 dx = 21x + C$$ So, the solution to the given integral is: $$\int \frac{21 x^{2}}{x^{3}-x^{2}-12 x} d x = 21x + C$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Long Division
Polynomial long division is a technique used to divide one polynomial by another. When the degree of the numerator is equal to or higher than the degree of the denominator, we may need to use this method. However, it's not always necessary if the division is straightforward.
In our example, we had \( \frac{21x^2}{x^3 - x^2 - 12x} \). The degrees of both parts seemed similar, but since the numerator was easily divisible by the denominator, long division showcased no remainder.
  • This means that simplification was possible without full division steps.
  • Recognizing this saves time and effort.
Polynomial long division can often look intimidating, but identifying opportunities to simplify before fully diving in is key.
Indefinite Integrals
Indefinite integrals represent a family of functions and are a core concept in calculus. An integral without bounds, called an indefinite integral, provides a general solution. It includes a constant \( C \) to account for any vertical shifts since differentiation of a constant results in zero.
In our task, we had \( \int 21 \, dx \), leading to \( 21x + C \).
  • The \( C \) is critical as it reflects all possible antiderivatives.
  • Indefinite integrals are closely related to the fundamental theorem of calculus, which links them with antiderivatives.
These capture the essence of integration as the reverse process of differentiation.
Integration Techniques
Integration isn鈥檛 always straightforward, so various techniques exist to handle different kinds of functions. These include substitution, integration by parts, and partial fractions. In our example, after simplifying through polynomial division, it became a simple constant integral \( \int 21 \, dx \).
Some key points about integration techniques include:
  • Always check for simplifications first鈥攖his can lead to easier integrals.
  • Understanding properties like linearity allows for breaking down complex problems.
  • Mastery of different techniques equips you for handling a broad range of calculus problems.
Using these approaches strategically helps in efficiently solving integrals.

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