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Chapter 4: Problem 66

Right, or wrong? Give a brief reason why. $$\int \frac{-15(x+3)^{2}}{(x-2)^{4}} d x=\left(\frac{x+3}{x-2}\right)^{3}+C$$

Short Answer

Expert verified
The expression is wrong because the integration does not simplify to the right side formula.

Step by step solution

01

Analyze the Given Expression

The given expression to differentiate is \(-15(x+3)^{2} / (x-2)^{4}\). This is a rational function where both the numerator and denominator are polynomials.
02

Consider Appropriate Integration Techniques

Given the rational function and its complexity, consider techniques like polynomial long division, substitution, or integration by parts. Notice that the structure might suggest a substitution.
03

Choose a Substitution

Attempt a substitution of \( u = \frac{x+3}{x-2} \) which simplifies components of the integral. Although replacing directly might not work easily, test if differentiating or manipulating could produce useful insights.
04

Revise Simplified Structure

Instead of complex substitution, check if simplifying the initial structure helps. Take any simpler derivative to test computations matching given integral. Note: this isn't a standard integral from the transformation, possibly illustrating direct derivatives don't match expected integral structure.
05

Derivative Confirmation

Derivatives confirm the validity. Since this structure implies comparing solutions or expressions, compute derivative of right side to match any potential integration of left. If mismatched, integration conclusion can be warranted incorrect.

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Key Concepts

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

Rational Functions
Rational functions are expressions that consist of the ratio of two polynomials. They are written in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). Because you can't divide by zero, rational functions have restrictions on their domain, usually determined by the points where the denominator \( Q(x) \) equals zero.
Understanding the behavior of rational functions involves analyzing:
  • Asymptotes: These are lines that the graph of the function approaches but never touches. A vertical asymptote occurs where the denominator is zero. Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials.
  • Holes: These occur at any common factors in the numerator and denominator if they cancel out, leaving a point where the function is undefined.
In integration, rational functions require careful consideration of algebraic manipulation, as they often don't fit neatly into basic integration formulas. Techniques like substitution and integration by parts may be necessary.
Substitution Method
The substitution method in integration, sometimes called 'u-substitution', is a technique used to simplify integrals. It involves changing variables to make an integral easier to solve. This method can be thought of as the reverse of the chain rule for differentiation.
To use the substitution method:
  • Identify a part of the integral to substitute with a new variable (often denoted as \( u \)). This part should be a function whose derivative also appears in the integral.
  • Rewrite the integral in terms of \( u \) and find \( du \), the differential of \( u \).
  • Substitute the new variable and its differential into the integral, transforming it into a simpler form.
  • Integrate with respect to \( u \).
  • Substitute back the original expression for \( u \) to get the final result.
In the given problem, an initial substitution \( u = \frac{x+3}{x-2} \) was suggested; however, simplifying the process further or testing other substitutions might be necessary to solve certain integrals effectively.
Integration by Parts
Integration by parts is a technique derived from the product rule for differentiation. It is useful for integrating the products of two functions. The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \] where \( u \) and \( dv \) are parts of the original integral. Choosing how to split the integral can significantly impact the complexity of solving the problem.
Steps in integration by parts:
  • Choose \( u \) and \( dv \) from the integral \( \int u \, dv \). Pick \( u \) such that its derivative \( du \) simplifies the integral.
  • Compute \( du \) and \( v \) by integrating \( dv \).
  • Substitute into the integration by parts formula.
  • Solve the resulting integrals.
This method is particularly helpful when other techniques don't easily apply. In rational functions like the exercise, deciding between substitution and integration by parts depends on which technique simplifies the integration most effectively.

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Most popular questions from this chapter

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