Problem 16 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log [f(x)]+C\), then \(f(x)\) is equal to (A) \(\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}\) (B) \(\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}\) (C) \(\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}\) (D) \(\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}\)

Step-by-Step Solution

Answer

The function \( f(x) \) is \( \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \).

Step 1: Recognize the integration form

The given integral is \( \int f(x) \sin x \cos x \, dx = \frac{1}{2(b^2 - a^2)} \log[f(x)] + C \). This suggests that derivative under the logarithm should match with the integrand.

Step 2: Differentiate the right side

Differentiate \( \frac{1}{2(b^2 - a^2)} \log[f(x)] \) with respect to \(x\), using the chain rule. The derivative is \( \frac{1}{2(b^2 - a^2)} \cdot \frac{f'(x)}{f(x)} \).

Step 3: Relate the differentiated function to the integrand

For the integral equation to hold, the derivative \( \frac{1}{2(b^2 - a^2)} \cdot \frac{f'(x)}{f(x)} \) must be equal to \( f(x) \sin x \cos x \). Thus, \( f(x) \cdot \frac{1}{f(x)}\cdot f'(x) \) should match with \( 2(b^2 - a^2) \sin x \cos x \).

Step 4: Set up the equation for function \( f(x) \)

Equating the above expression to the integrand gives us \( f'(x) = 2(b^2 - a^2) \sin x \cos x \cdot f(x)^2 \). Rearranging this provides the differential equation for \( f(x) \): \( \frac{f'(x)}{f(x)^2} = 2(b^2 - a^2) \sin x \cos x \).

Step 5: Identify \( f(x) \) using pattern matching

Recognizing the structure of \( f(x) \), compare \( f(x) \sin x \cos x \) with potential candidates. Check which candidate fits the equation form of \( f'(x) \) and revert to simplest function matching the integration pattern.

Step 6: Choose the correct option

By observation and substitution, verify that \( f(x) = \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \) satisfies all derived conditions and matches the integral setup.

Key Concepts

Differential Equations

Differential equations play a crucial role in understanding the dynamics of complex systems that change with respect to one or more variables. In this exercise, a differential equation is formed through integration and differentiation processes, providing insight into how a function changes. Here, after differentiating the logarithmic expression, we derived: \[\frac{f'(x)}{f(x)^2} = 2(b^2 - a^2) \sin x \cos x.\]

• This shows a relationship between the derivative of the function \(f(x)\) and the variables \(\sin x\) and \(\cos x\).
• Solving this differential equation involves recognizing a pattern or a structure that simplifies the function \(f(x)\).

Solving such equations often requires techniques like separation of variables, integrating factors, or substitution, depending on the structure. In our exercise, identifying the structure quickly points us to a known function form, possibly avoiding more complex computations. Always remember that differential equations are vital for modeling a wide range of phenomena in physics, engineering, biology, and economics; here, they guide us toward identifying the structure of \(f(x)\).

Trigonometric Functions

Trigonometric functions like \(\sin x\) and \(\cos x\) are not just simple periodic functions but tools that model oscillatory behavior, wave functions, and many natural phenomena. In this exercise:

• We observe these functions in the integrand \(f(x) \sin x \cos x\), bringing in their identities and properties.
• The product \(\sin x \cos x\) can also be expressed using trigonometric identities, potentially simplifying calculations. It can be rewritten as \(\frac{1}{2}\sin(2x)\), a transformation gain for integration or differentiation techniques.

Understanding their properties helps in evaluating integrals or identifying patterns within equations. This exercise primarily utilizes these functions to shape \(f(x)\), indicating their versatility beyond just simple computation. Whether they help in simplifying integrands or solving derivatives, trigonometric functions are widely applicable in pure and applied mathematics.

Logarithmic Differentiation

Logarithmic differentiation is a technique used when differentiating functions that are products, quotients, or have variable exponents. In this context:

• We differentiate the expression \(\frac{1}{2(b^2 - a^2)} \log[f(x)]\) using the chain rule.
• This involves taking the derivative of the logarithm, which is \(\frac{1}{f(x)} f'(x)\), and then factoring in the constant multiplier.

This process simplifies the differentiation of more complex functions by converting multiplicative relationships into additive ones via logarithms. Such transformation makes it easier to differentiate each component individually. Logarithmic differentiation can be especially useful when handling complex exponentials or products involving several functions. It is a dynamic method to reduce complexity and solve intricate derivative problems efficiently, as observed in the exercise.