Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

Column-I Column-II (A) \(\int \sqrt{\sec x-1} d x\) 1\. \(\sin ^{-1}(\tan x)\) (B) \(\int \frac{d x}{\cos x \sqrt{\cos 2 x}}\) 2\. \(\sec ^{-1}(\sec x+\cos x)\) (C) \(\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}\) 3\. \(-2 \log \left(\cos \frac{x}{2}+\sqrt{\cos ^{2} \frac{x}{2}-\frac{1}{2}}\right)\) (D) \(\int \frac{\sin ^{3} x d x}{\left(1+\cos ^{2} x\right) \sqrt{1+\cos ^{2} x+\cos ^{4} x}}\) 4\. \(\sqrt{2}\left(\sqrt{\tan x}+\frac{1}{5} \tan ^{5 / 2} x\right)\)

Step-by-Step Solution

Answer

(A)-4, (B)-1, (C)-3, (D)-2.

Step 1: Match the First Integral

We start with column (A) integrand \(\int \sqrt{\sec x-1} \, dx\). Given options in Column-II, we interpret the structure to match potential transformations. We will match this with (D) by recognizing it involves transformations similar to option 4, particularly involving \(\tan x\) and its powers. Thus, match: (A)-4.

Step 2: Simplify the Second Integral

For column (B), \(\int \frac{dx}{\cos x \sqrt{\cos 2x}}\), we notice that changes of variables or identities involving \(\cos 2x\) could lead to direct expressions similar to inverse trigonometric forms. This aligns with option 1, involving \(\sin^{-1}(\tan x)\) . Hence, match: (B)-1.

Step 3: Evaluate the Third Integral

Analyze column (C), \(\int \frac{dx}{\cos^3 x \sqrt{\sin 2x}}\). This requires trigonometric transformations that typically express in logarithmic format. Option 3 suits the expression when simplifying factors using \(\log(x)\) functions and trigonometric identities. Therefore, match: (C)-3.

Step 4: Determine the Fourth Integral Structure

Finally, column (D) with integrand \(\int \frac{\sin^3 x \, dx}{(1+\cos^2 x) \sqrt{1+\cos^2 x+\cos^4 x}}\) often fits complex transformations, found in secant inverse functions like option 2 \(\sec^{-1}(\sec x+\cos x)\). Thus, match: (D)-2.

Key Concepts

Trigonometric Integrals

Trigonometric integrals are foundational in calculus, involving integrals that include trigonometric functions such as sine, cosine, tangent, and others. These integrals often require clever substitutions and identities for simplification. For example, integrating expressions like \( \int \sin^n(x) \cos^m(x) \, dx \) typically changes form using identities such as \( \sin^2(x) + \cos^2(x) = 1 \).

When tackling trigonometric integrals:

• Identify symmetries in sine and cosine functions.
• Use substitution effectively, such as \( u = \sin(x) \) or \( u = \cos(x) \), which simplify integrals by reducing their complexity.
• Consider using trigonometric identities for reduction, like transforming \( \sin(2x) \) into \( 2 \sin(x)\cos(x) \). These can powerfully transform a difficult integral into something more manageable.

Mastering these techniques, such as in the provided exercise, allows you to handle complex integrals that combine products and powers of trigonometric functions with ease.

Definite and Indefinite Integration

In calculus, understanding the differences between definite and indefinite integrals is essential. An indefinite integral represents a family of functions and includes a constant of integration, \( C \). It's expressed in the form:

• \( \int f(x) \, dx = F(x) + C \)

Meanwhile, a definite integral calculates the area under the curve of \( f(x) \) from \( x = a \) to \( x = b \), and it's written as follows:

• \( \int_{a}^{b} f(x) \, dx \)

Upon evaluation, the result is a single numerical value, rather than a family of functions.

Some useful tips for solving these integrals include:

• For indefinite integrals, always remember to add the constant of integration, as it accounts for any arbitrary vertical shifts in the graph of the antiderivative.
• Apply the Fundamental Theorem of Calculus to evaluate definite integrals, which connects differentiation and integration in a powerful way.
• Be vigilant with integration endpoints and any discontinuities within the interval when working with definite integrals. This helps in avoiding any oversight that might result in incorrect calculations.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \), are the inverse operations of standard trigonometric functions. They are used to find angles with a known trigonometric value. For instance, if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).

These functions are important in integration as they help solve integrals involving square roots and expressions that have trigonometric forms. Some important aspects include:

• Recognize typical forms, like the integral \( \int \frac{dx}{\sqrt{1-x^2}} \), which corresponds to \( \sin^{-1}(x) + C \).
• They are useful for solving integrals that appear in the format of \( \int \frac{1}{1+x^2} \, dx \), which leads to \( \tan^{-1}(x) + C \).
• Inverse trigonometric functions often appear when tasked with finding areas or computing certain limits involving trigonometric expressions.

The integration task often involves recognizing these forms to simplify the calculation process. Mastery of these functions can make solving seemingly complex integrals quite straightforward.