Problem 7 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

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

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{ac} \tan^{-1} \frac{ax+b}{c} + C \).

Step 1: Recognize the Integral Type

The integral \( \int \frac{d x}{c^{2}+(a x+b)^{2}} \) is of the form \( \int \frac{d x}{A^2 + u^2} \), which is known to be the inverse tangent integrals, \( \frac{1}{A} \tan^{-1} \frac{u}{A} + C \).

Step 2: Identify Parameters

In our integral, \( A=c \) and \( u=ax+b \). This allows us to match the form of the known inverse tangent integral.

Step 3: Apply Substitution

Set \( u = ax + b \), making \( d u = a\,d x \), or \( d x = \frac{1}{a} \, d u \). Substitute into the integral:\[\int \frac{1}{c^2 + u^2} \frac{1}{a} \, d u = \frac{1}{a} \int \frac{1}{c^2 + u^2} \, d u.\]

Step 4: Solve the Integral

Using the formula \( \int \frac{1}{c^2 + u^2} \, d u = \frac{1}{c} \tan^{-1} \frac{u}{c} + C \), we find:\[\frac{1}{a} \left( \frac{1}{c} \tan^{-1} \frac{u}{c} \right) + C = \frac{1}{ac} \tan^{-1} \frac{u}{c} + C.\]

Step 5: Substitute Back

Since \( u = ax + b \), substitute back:\[\frac{1}{ac} \tan^{-1} \frac{ax+b}{c} + C. \]

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus that involves finding the antiderivative or area under a curve represented by a function. There are several techniques to approach integration, especially when dealing with complex functions. Here are some of the most common techniques:

• Direct Integration: This is suitable for basic algebraic functions where the antiderivative is easily identifiable.
• Substitution Method: Often used to simplify the integrand by changing variables, making it easier to integrate.
• Integration by Parts: Useful when integrating products of functions by breaking them into parts.
• Partial Fraction Decomposition: Specifically helpful when dealing with rational functions, by expressing them as a sum of simpler fractions.

In our problem, the integral may seem daunting at first as it involves a rational function of the form \( c^2 + (ax+b)^2 \). However, with the right technique like substitution, this integral can be simplified neatly into a well-known inverse tangent integral form.

Substitution Method

The substitution method is a powerful tool in integration, especially when you encounter an integral that doesn't seem straightforward. It transforms a complex function into an easier one by replacing variables.

Here's how it works:

• Identify part of the integrand as a new variable, say \( u \). In our example, we let \( u = ax + b \).
• Differentiate the new variable to express \( dx \) in terms of \( du \). For instance, \( du = a \, dx \), thus \( dx = \frac{1}{a} \, du \).
• Replace all occurrences of the old variable with the new one. Our integral \( \int \frac{dx}{c^2 + (ax+b)^2} \) becomes \( \frac{1}{a} \int \frac{du}{c^2 + u^2} \).

This substitution simplifies the integration process significantly. The new integral is a standard form that can be easily evaluated, bypassing more complex calculations.

After solving the integral in terms of \( u \), we substitute back to express the solution in terms of the original variable \( x \). The substitution method not only simplifies calculations but also enhances understanding of the integral's structure.

Integration of Rational Functions

Rational functions are expressions that involve a ratio of polynomials, and integrating these can sometimes be tricky. However, there are methods to simplify and solve these integrals effectively.

Firstly, observe the structure of our given integral, \( \int \frac{dx}{c^2 + (ax+b)^2} \), which is a rational function due to the polynomial \( c^2 + (ax+b)^2 \) in the denominator. Rational functions can be tackled through various techniques:

• Using trigonometric relationships to simplify, especially if the expression suggests forms related to trigonometric identities.
• Decomposing complex fractions into simpler parts with Partial Fraction Decomposition (though not applied here, it is useful in other contexts).
• Recognizing special integral forms such as the inverse tangent integral, which arises frequently in rational expressions of certain types.

In this specific exercise, recognizing that the expression fits the inverse tangent integral form, \( \frac{1}{A^2 + u^2} \), we can apply the recognized solution directly. This turns the seemingly complex task of integrating a rational function into a straightforward application of a known formula, allowing quicker and accurate calculations.