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Chapter 1: Problem 12

\(\int \frac{a d-b c}{(c x+d)^{2}} d x=\frac{a x+b}{c x+d}\)

Short Answer

Expert verified
The integral \(\int \frac{a d-b c}{(c x+d)^{2}} \, d x\) simplifies to \(-\frac{(a d - b c)}{c (c x + d)} + C.\)

Step by step solution

01

Understand the Given Problem

We are given an integral to solve: \(\int \frac{a d-b c}{(c x+d)^{2}} \, d x = \frac{a x+b}{c x+d}\). The task is to verify or prove the equality of the integral and the expression on the right side.
02

Integrate the Left Side

To solve the integral \(\int \frac{a d-b c}{(c x+d)^{2}} \, d x\), recognize that this integral can be evaluated as a constant times the derivative of \(\frac{1}{c x + d}\). The integral of \(\frac{1}{(c x + d)^{2}}\) is \(-\frac{1}{c} \cdot \frac{1}{c x + d}\). Thus, we have:\[\int \frac{a d-b c}{(c x+d)^{2}} \, d x = -(a d-b c) \cdot \frac{1}{c} \cdot \frac{1}{c x + d} + C\].
03

Simplify the Expression

The expression becomes:\[-\frac{(a d - b c)}{c (c x + d)} + C.\]

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Key Concepts

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

Integration Techniques
Integration techniques are tools we use to solve integrals, which can vary in difficulty from very simple to extremely complex.
Common methods include substitution, integration by parts, partial fraction decomposition, and more advanced techniques like trigonometric substitutions. For this exercise, we focus on recognizing a special form of integral and using knowledge about derivatives.
  • Identify Patterns: Recognize if the integral matches a known pattern or a derivative formula. This helps to simplify the integration process.
  • Choose the Right Technique: For example, when faced with \(\int \frac{a d-b c}{(c x+d)^{2}} \, d x\), we note that this matches the derivative of \(-\frac{1}{c} \cdot \frac{1}{c x + d}\).

Mastering these techniques requires practice, and understanding the fundamental rules and relationships in calculus.
Definite and Indefinite Integrals
Integrals can be classified into definite and indefinite forms.
  • Indefinite Integrals: These represent a family of functions and include an arbitrary constant, \(C\), because many antiderivatives can result in the same derivative.
  • Definite Integrals: These yield a specific numerical value representing the area under a curve between two bounds.
In the provided exercise, the integral is indefinite as there are no bounds given, and the constant \(C\) is included.
The indefinite integral solves to the left-hand expression as a function of \(x\), with terms simplified to the form \(-\frac{(a d - b c)}{c (c x + d)} + C\), revealing the antiderivative.
When understanding these integrals, consider that while definite integrals calculate actual quantities, indefinite ones give us functions that describe a family of solutions.
Rational Functions Integration
Rational functions are ratios of polynomials and often appear in calculus problems.
Integration of rational functions can be tricky, often involving techniques like partial fraction decomposition or functional recognition.
  • Recognizing Formats: Noticing forms like \(\frac{1}{(c x + d)^2}\), which frequently shows up, can speed up the integration process as we turn to known derivatives.
  • Simplifying Given Terms: Waveward simplifies our \( (a d - b c) \) over \( (c x + d)^2 \) into manageable parts with the right technique.
In the exercise, direct recognition helps: understanding that the integral simplifies took advantage of known derivative patterns.
Practicing with various examples of rational function integration will make this increasingly intuitive.

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Most popular questions from this chapter

$$ \int \frac{d x}{a^{2}-x^{2}}=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right| \text { or } \frac{1}{a} \tan h^{-1}\left(\frac{x}{a}\right) $$

Lvaluate \(\int \frac{1}{(x+1) \sqrt{x^{3}-1}} d x\)

Fvaluate \(\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x\)

$$ \int \frac{d t}{\sqrt{(t+\alpha)(t+\beta)}}=2 \log [\sqrt{t+\alpha}+\sqrt{t+\beta}] $$

Lvaluate \(\int \sqrt{\frac{a+x}{x}} d x\)
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