Problem 12 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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:\[\int \frac{a d-b c}{(c x+d)^{2}} \, d x = -(a d-b c) \cdot \frac{1}{c} \cdot \frac{1}{c x + d} + C\].

Step 3: Simplify the Expression

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

Key Concepts

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.