Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x}=\) (A) \(\frac{1}{2}(\sin x-\cos x+x)+c\) (B) \(\frac{1}{2}(\sin x-\cos x-x)+c\) (C) \(\frac{1}{2}(\sin x+\cos x+x)+c\) (D) none of these

Step-by-Step Solution

Answer

(D) none of these

Step 1: Simplify the integrand

Begin by simplifying the expression \( \tan x + \cot x + \sec x + \csc x \). This can be rewritten using equivalent trigonometric functions:\[ \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}, \quad \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x} \]. Combine these under a common denominator, which is \( \sin x \cos x \) and simplify:\[ \frac{\sin^2 x + \cos^2 x + 1}{\sin x \cos x} = \frac{2}{\sin x \cos x} = \frac{2}{\frac{1}{2}\sin(2x)} = \frac{4}{\sin(2x)} \].

Step 2: Rewrite using a known identity

Recognize the simplified form as \( \frac{4}{\sin(2x)} \) which can be rewritten using the identity for \( \sin(2x) \). This becomes \[ \frac{4}{2 \sin x \cos x} = \frac{2}{\sin x \cos x} \].Check for further simplifications or alternative forms, but now it is clear \( \frac{4}{\sin(2x)} \) simplifies to this form.

Step 3: Integrate the simplified function

The integrand simplifies to \( \int \frac{4}{\sin(2x)} \, dx \), which can be represented as \( 2 \cdot \int \csc(2x) \, dx \). Using the standard integral \( \int \csc x \, dx = -\ln|\csc x + \cot x| + C \), substitute:\[ 2 \cdot \left( -\frac{1}{2}\ln |\csc(2x) + \cot(2x)| \right) = -\ln |\csc(2x) + \cot(2x)| + C \].

Step 4: Analyze result vs options

Our integral \( -\ln |\csc(2x) + \cot(2x)| + C \) does not match any of the provided options, thus the correct answer is that none of the given options are equivalent. Therefore, the answer to the problem is (D) none of these.

Key Concepts

Trigonometric Integration

Trigonometric integration is a fundamental process in calculus, specifically valuable when dealing with integrals that involve trigonometric functions. This method often involves breaking down complex trigonometric expressions into more basic components, making them easier to integrate. When tackling a problem involving trigonometric integration, it's crucial to recognize the trigonometric functions involved such as \( \tan x \), \( \cot x \), \( \sec x \), and \( \csc x \). Each of these can be rewritten:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)

Breaking down expressions using these basic trigonometric identities can lead to simpler forms that are more straightforward to integrate. Ultimately, this simplifies complex integrals and makes the integration process much more manageable. Understanding these identities and how to apply them in an integration context is key to mastering trigonometric integration.

Integration Techniques

Integration techniques are essential tools in calculus that help solve integral problems that are not straightforward. In this exercise, we use a specific approach known as 'trigonometric substitution' and manipulate the function so it becomes easier to integrate. A common tactic is to simplify the problem by substituting trigonometric identities and algebraic manipulations. For instance, simplifying an expression by finding a common denominator can turn a complex trigonometric expression into something more manageable. As seen in the solution, combining terms under a common denominator allowed the integrand to be rewritten as \( \frac{4}{\sin(2x)} \), which could then be related to a familiar trigonometric identity.

• Common denominator aids in simplification.
• Identification of substitution helps in uncovering simpler integral forms.

By using these techniques, it's possible to convert daunting integrands into simpler forms, which then can often be integrated using standard integral results like \( \int \csc x \, dx \), leading to solutions that may not be immediately obvious.

Trigonometric Identities

Trigonometric identities are expressions involving trigonometric functions that are true for every value of the occurring variables. These identities are critical tools when performing integration, as they allow for the transformation of complex trigonometric expressions into simpler, more operable forms. Key identities utilized in solving integrals include:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( \sin(2x) = 2\sin x \cos x \)

In the given exercise, the equation \( \sin^2 x + \cos^2 x + 1 \) simplifies using these trigonometric identities, making the integration process feasible. Recognizing and applying these identities in the context of integrals allows for the decomposition of terms into simpler trigonometric functions, which are easily relatable to basic integrals.This problem illustrates how essential trigonometric identities are in calculus, specifically in reducing integrals to a form that can be tackled using standard integration techniques.