Problem 58 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{\left(1-\cot ^{n-2} x\right) d x}{\tan x+\cot x \cdot \cot ^{n-2} x}=\) (A) \(\frac{1}{n} \log \left|\sin ^{n} x-\cos ^{n} x\right|+c\) (B) \(\frac{1}{n} \log \left|\sin ^{n} x+\cos ^{n} x\right|+c\) (C) \(\frac{1}{n-1} \log \left|\sin ^{n} x+\cos ^{n} x\right|+c\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A) \(\frac{1}{n} \log \left|\sin ^{n} x-\cos ^{n} x\right|+c\).

Step 1: Rewrite the Integrand

Notice that we have both \( \tan x \) and \( \cot x \) functions in the integrand. We can rewrite \( \tan x \) as \( \frac{\sin x}{\cos x} \) and \( \cot x \) as \( \frac{\cos x}{\sin x} \). This allows us to express the integrand entirely in terms of sine and cosine functions, which can simplify integration.

Step 2: Substitution Simplification

Make the substitution \( u = \cot^{n-2} x \), then \( du = -(n-2) \cot^{n-3} x \csc^2 x \, dx \). Also, rewrite \( \tan x = \frac{1}{\cot x} \) in terms of \( u \). The integral simplifies substantially under this substitution.

Step 3: Reformat the Integral

After substitution, the integral becomes a function in terms of \( u \). Simplifying the expression, the integral can be rewritten to help identify a possible antiderivative. Use algebra to manage the terms after substitution, focusing on forming simpler, recognizable integral forms.

Step 4: Identify the Integral Structure

Recognize the transformed integral as a derivative of a logarithmic function considering the form inside the integral and the connection to the possible solutions given in options. Align the structure of the integral with the standard integral formulas.

Step 5: Evaluate the Integral

Using logarithmic integration rules, evaluate the integral by recognizing the derivative of \( \log \) functions that match the structure of the integral from earlier manipulations.

Step 6: Consider Constants and Final Form

After integrating, add the constant of integration \( c \) to your result. Make sure the result aligns with one of the provided options by comparing the integral result. The option that matches the form derived from the integration correctly should be selected as the final answer.

Key Concepts

Trigonometric Integration

When dealing with trigonometric integration, especially with functions involving both \(\tan x\) and \(\cot x\), expressing these functions in terms of sine and cosine is often very helpful. This is because many trigonometric identities and simplifications can be applied to sine and cosine, making the integrals easier to evaluate.

• \(\tan x\) can be rewritten as \(\frac{\sin x}{\cos x}\).
• \(\cot x\) can be expressed as \(\frac{\cos x}{\sin x}\).

Maintaining a focus on these identities allows us to transform the integrals into a form that can be easily tackled by techniques like substitution or by recognizing standard integral forms. This often aids in simplifying complex expressions into basic logarithmic or algebraic forms, upon which integration becomes more straightforward.

Substitution Method

The substitution method is a powerful tool for simplifying integrals, particularly when the integrand is a composition of functions. Here, we specifically use substitution to tackle the complexity of combinations like \( \tan x \) and \( \cot x \).
Substitution steps might look like this:

• Choose a substitution that simplifies the integral. In the case above, substituting \( u = \cot^{n-2} x \) simplifies the problem.
• Derive \( du \) from the chosen substitution. This requires computing \( du = -(n-2) \cot^{n-3} x \csc^2 x \, dx \).
• Replace the expressions in the integrand with your \( u \) and \( du \) terms.

By doing this, the integral initially clouded with complex trigonometric identities, clears up to a more manageable form, focusing on a single variable \( u \), thereby simplifying the integration process greatly.

Logarithmic Integration

Logarithmic integration is an integral technique essential for solving problems that result in logarithmic functions after manipulation. This is especially useful when the integral transforms into a function of \( \ln |f(x)| \) as part of its solution.
In the solution provided, notice how the integral is eventually evaluated by recognizing the form of a logarithmic derivative:

• Identify the integral's structure to see if it aligns with \( \frac{1}{f(x)} \frac{df(x)}{dx} = \frac{d}{dx} \ln |f(x)| \).
• Once in this recognized form, integrate to yield \( \ln |f(x)| \).
• Incorporate constants of integration and compare your result with provided solution options to check for alignment.

This technique underscores how recognizing derivative forms directly simplifies the integration process by projecting it to known function derivative relationships, such as the derivative of a logarithm.