Problem 4 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int \frac{d x}{(a x+b)^{2}}=\frac{-1}{a(a x+b)}\)

Step-by-Step Solution

Answer

The integral is \( \frac{-1}{a(ax+b)} + C \).

Step 1: Identify the Function to Integrate

The function we need to integrate is \( \frac{1}{(ax+b)^2} \). This is a rational function, and can be solved using the method of substitution, which is suitable for dealing with expressions of this form.

Step 2: Choose an Appropriate Substitution

Let's choose the substitution \( u = ax + b \). This choice simplifies the expression within the integral. Differentiating \( u \) with respect to \( x \) gives \( du = a \, dx \), or \( dx = \frac{du}{a} \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) and \( dx \) into the integral. Thus, the integral becomes \[ \int \frac{1}{u^2} \cdot \frac{du}{a} = \frac{1}{a} \int \frac{1}{u^2} \, du. \]

Step 4: Integrate the Simplified Expression

The integral \( \int \frac{1}{u^2} \, du \) is a standard integral that results in \( -\frac{1}{u} \). So, we have \[ \frac{1}{a} \left( -\frac{1}{u} \right) = -\frac{1}{au}. \]

Step 5: Substitute Back in Terms of x

Replace \( u \) with \( ax + b \) to express the result in terms of \( x \). This gives \( -\frac{1}{a(ax+b)} \).

Step 6: State the Final Result

The integral of \( \frac{1}{(ax+b)^2} \) with respect to \( x \) is \( \frac{-1}{a(ax+b)} + C \), where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify integration, especially when dealing with more complex functions. It involves replacing a more complex expression within the integral with a simpler variable, usually denoted as \( u \). The goal is to transform the original integral into a simpler form that is easier to integrate. To achieve this, follow these steps:

• Identify a substitution that simplifies the integral. In our example, we used \( u = ax + b \).
• Differentiate \( u \) to find \( du \) and express \( dx \) in terms of \( du \): \( dx = \frac{du}{a} \).
• Substitute \( u \) and \( dx \) back into the integral.

Using these steps, the original integral becomes simpler and more manageable, allowing you to solve it more efficiently.

Rational Functions

Rational functions are quotients of two polynomials. In our exercise, the function \( \frac{1}{(ax+b)^2} \) is a rational function, since it can be expressed as the quotient of the constant 1 over a polynomial \((ax+b)^2\).
When integrating rational functions, several techniques can be employed, such as:

• Substitution: Perfect for transforming polynomials and making the integral simpler.
• Partial Fraction Decomposition: Useful when dealing with more complex polynomials.
• Direct Integration: Applicable when the rational function is already in a simple form.

For the integral \( \int \frac{1}{(ax+b)^2} \, dx \), substitution is the most straightforward method, making the expression easy to integrate.

Definite and Indefinite Integrals

In calculus, integrals are classified into two main types: definite and indefinite integrals. Our exercise focuses on indefinite integrals, where we find an antiderivative of a function without specifying the limits of integration.

• Indefinite Integrals: Represent the family of all antiderivatives of a function. The result always includes a constant \( C \), as seen in \( \frac{-1}{a(ax+b)} + C \).
• Definite Integrals: Calculate the area under a curve between two limits. The result is a number and no arbitrary constant is included.

Understanding the distinction between these two is crucial for applying them correctly in problems. In cases where the integral bounds are given, the integration would be definite, providing the exact magnitude of the area under the curve.