Problem 188 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\frac{d}{d x} F(x)=\left(\frac{e^{\sin x}}{x}\right), x>0\). If \(\int_{1}^{4} \frac{3}{x} e^{\sin x^{t}} d x=F(k)-F(1)\), then one of the possi- ble values of \(k\), is (A) 15 (B) 16 (C) 63 (D) 64

Step-by-Step Solution

Answer

The possible value of k is (A) 15.

Step 1: Understand the Exercise

We are given that the derivative of a function \( F(x) \) is \( \left(\frac{e^{\sin x}}{x}\right) \), and we need to evaluate the integral \( \int_{1}^{4} \frac{3}{x} e^{\sin x^{t}} dx \) in terms of \( F(k) - F(1) \). Our task is to find a possible value for \( k \).

Step 2: Simplify the Integral Expression

Rewriting \( \int_{1}^{4} \frac{3}{x} e^{\sin x^{t}} dx \) in terms of \( F(x) \), observe that it likely represents a transformation related to the derivative \( \frac{d}{dx} F(x) = \frac{e^{\sin x}}{x} \). However, the integral is \( \int \frac{3}{x} e^{\sin x^{t}} dx \), suggesting a higher power or transformation involving constants. Further evaluation suggests symmetry or scaling.

Step 3: Reinterpret Integral in Terms of F(x)

The given function \( F(x) \) isn't directly indicated in the integral, but because it involves an expression \( \int \frac{3}{x} e^{\sin x} \), it’s evident that we might need to scale or adjust the integrand to reflect the scaled condition involving the constant multiple factor of 3 or consider transformed or nested integrals.

Step 4: Evaluate the Possible Values of k

From the given options, match assumptions for transformations (such as chain rule or composite functions), and assume resulting conditions to cause entire multiplication adjustments or function composition regarding values such as 15, 16, 63, 64 in context derivation of \( (F(x))^t \sim 3F(1)\): distribution or presentation principles.

Step 5: Conclusion for the Value of k

Substituting values such as \( t = 2 \) and an approximation evaluation for F(k) \(-3F(1)\), leads towards hypothesis check implying within given methods, the correctness inclines towards option (A) 15.

Key Concepts

Derivatives

In calculus, derivatives measure how a function changes as its input changes. It's like looking at the slope of a curve at any given point. For instance, if you have a function representing the path of a car, the derivative tells you the car's speed at each point in time.

To find the derivative of a given function, like in our exercise, you might encounter expressions involving exponentials or trigonometric functions. The derivative \(\frac{d}{dx} F(x) = \frac{e^{\sin x}}{x}\) occurs frequently in complex functions and requires understanding specific rules such as:

• Product Rule: Used when differentiating products of two functions.
• Quotient Rule: Useful for derivatives of fractions where both numerator and denominator are variables.
• Chain Rule: Helps when the function is composed of another function, such as trigonometric or exponential parts.

The derivative gives us instant rates of change, essential for graphing functions and solving real-world problems.

Integrals

Integrals are the opposite operation of derivatives. They accumulate quantities and can be thought of as finding the total area under a curve. The problem gives an expression \(\int_1^4 \frac{3}{x} e^{\sin x^{t}} \ dx\)eeding evaluation in terms of a difference of two integrals.To tackle integrals, especially when dealing with more complex expressions, familiarize yourself with integral properties:

• Definite Integrals: These calculate a precise area, with specified bounds like 1 and 4 in example.
• Power Rule: Pertains to integrating functions raised to a power.
• Substitution Method: Useful in simplifying integrals by changing variables to simplify the function, making integration easier.

In our exercise, the function inside the integral may require scaling. This reveals composite or transformed functions, often necessitating advanced techniques or approximations for precise values.

Function Transformation

Transforming a function involves altering its overall form or structure, like shifting, stretching, or rotating it on a graph. In the problem, function transformation arises as we need to reinterpret how the function appears under a different guise or more complex form, perhaps due to exponentiation or scaling by a factor.

Several transformations can alter functions effectively:

• Vertical Shifts: Moving a graph up or down by adding or subtracting a constant from the function.
• Horizontal Shifts: Moving a graph left or right requires modifying the input variable by a constant.
• Stretching/Compressing: Achieved by multiplying the function by a scale factor.

In calculus problems, understanding these transformations is key because they can drastically simplify solving integration and differentiation. Modifying expressions to better fit standard patterns can unravel complex nested integrals or composites, as evident in the given exercise.