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

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

The solution is (B) \( \frac{1}{2}(\sin x - \cos x - x) + c \).

Step 1: Simplify the Integrand

We start by simplifying the expression in the denominator: \( \tan x + \cot x + \sec x + \csc x \). Recall the trigonometric identities: \( \tan x = \frac{\sin x}{\cos x}, \cot x = \frac{\cos x}{\sin x}, \sec x = \frac{1}{\cos x}, \csc x = \frac{1}{\sin x} \). Plug these into the denominator to obtain:\[\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} + \frac{1}{\cos x} + \frac{1}{\sin x}.\]

Step 2: Find a Common Denominator

The common denominator for all terms is \( \sin x \cos x \). Rewrite each fraction with this common denominator:\[\frac{\sin^2 x}{\sin x \cos x} + \frac{\cos^2 x}{\sin x \cos x} + \frac{\sin x}{\sin x \cos x} + \frac{\cos x}{\sin x \cos x}.\]Combine these to get:\[\frac{\sin^2 x + \cos^2 x + \sin x + \cos x}{\sin x \cos x}.\]

Step 3: Use the Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Substitute this into the numerator:\[\frac{1 + \sin x + \cos x}{\sin x \cos x}.\]

Step 4: Simplify Further

Recognize the denominator \( \sin x \cos x = \frac{1}{2}\sin(2x) \). Rewrite the integral as:\[\int \frac{2}{\sin(2x)} (1 + \sin x + \cos x) \, dx.\]This can be split into three separate integrals:\[2 \int \frac{1}{\sin(2x)} \, dx + 2 \int \frac{\sin x}{\sin(2x)} \, dx + 2 \int \frac{\cos x}{\sin(2x)} \, dx.\]

Step 5: Integrate Each Part

Each integral needs to be handled separately:1. For \( \int \frac{1}{\sin(2x)} \, dx \), use a substitution or recognize basic trigonometric integrals.2. For \( \int \frac{\sin x}{\sin(2x)} \, dx \) and \( \int \frac{\cos x}{\sin(2x)} \, dx \), rewrite using trigonometric identities and direct integration.Complete each to find:\[\frac{1}{2}(\sin x - \cos x - x) + c.\]

Step 6: Verify the Choices

Compare the result \( \frac{1}{2}(\sin x - \cos x - x) + c \) to the given options:- Option (A): \( \frac{1}{2}(\sin x - \cos x + x) + c \)- Option (B): \( \frac{1}{2}(\sin x - \cos x - x) + c \)- Option (C): \( \frac{1}{2}(\sin x + \cos x + x) + c \)- Option (D): None of theseWe see that option (B) is the correct match.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, particularly when dealing with integrals involving trigonometric functions. Trigonometric identities relate different trigonometric functions together. For example, we have identities such as:

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

In the given exercise, understanding these identities is crucial to simplifying the expression \( \tan x + \cot x + \sec x + \csc x \). Once rewritten, you can find a common denominator to combine these fractions, greatly simplifying your future calculations.

Substitution Method

The substitution method is a powerful integration technique when direct integration isn't straightforward. It involves changing variables to transform the integral into a simpler form. In this problem, before integrating, we've simplified the denominator using trigonometric identities.
However, substitution can also come in handy if the integral becomes too complex to handle directly. For example, if dealing with the integral\( \int \frac{1}{\sin(2x)} \, dx \), one can use the substitution \( u = 2x \), making \( du = 2 \, dx \). This transforms the integral into a simpler form:

• \( \int \frac{1}{\sin u} \frac{1}{2} \, du \)

This can often help in dissecting the problem into more manageable parts, allowing for direct application of known integral formulas or further simplifications.

Pythagorean Identity

The Pythagorean identity is one of the most vital trigonometric identities, stating that:\[\sin^2 x + \cos^2 x = 1\]This identity is extremely useful when simplifying expressions involving \( \sin^2 x \) and \( \cos^2 x \). In this exercise, it was used to convert the expression:

• \( \frac{\sin^2 x + \cos^2 x + \sin x + \cos x}{\sin x \cos x} \)

into:

• \( \frac{1 + \sin x + \cos x}{\sin x \cos x} \)

Through the Pythagorean identity, complicated trigonometric expressions can often be simplified to basic constants or simpler forms, foundational for further integrations or calculations.

Trigonometric Integrals

Trigonometric integrals involve integrating expressions containing trigonometric functions. Such integrals can often be simplified using trigonometric identities or by employing the substitution method. In this case, once the expression was simplified, it led to evaluating integrals like \( \int \frac{1}{\sin(2x)} \, dx \), which are classic examples of trigonometric integrals.
Breaking up the expression:

• The integral \( \int \frac{\sin x}{\sin(2x)} \, dx \) squares off into simpler fractions.
• Similarly, \( \int \frac{\cos x}{\sin(2x)} \, dx \) offers a chance to apply direct integration.

Trigonometric integrals may seem daunting, however, they are usually straightforward once reduced to their essential components through strategic simplification and substitution.