Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\int_{0}^{y} e^{-t^{2}} d t+\int_{0}^{x^{2}} \sin ^{2} t d t=0\), then \(\frac{d y}{d x}\) is equal to (A) \(2 x \sin ^{2} x^{2} e^{y^{2}}\) (B) \(-2 x \sin ^{2} x^{2} e^{y^{2}}\) (C) \(x \sin ^{2} x^{2} e^{y^{2}}\) (D) \(-x \sin ^{2} x^{2} e^{y^{2}}\)

Step-by-Step Solution

Answer

The correct answer is (B) \(-2 x \sin^2 x^{2} e^{y^{2}}\).

Step 1: Understand the Problem

We are given an equation involving two definite integrals and must find the derivative \( \frac{dy}{dx} \). The equation is: \( \int_{0}^{y} e^{-t^{2}} dt + \int_{0}^{x^{2}} \sin^{2} t dt = 0 \).

Step 2: Differentiate the Equation Implicitly

Differentiate both sides of the equation with respect to \( x \). Use the Fundamental Theorem of Calculus, which states that \( \frac{d}{dx} \int_{a}^{u(x)} f(t) dt = f(u(x)) \frac{du}{dx} \). This yields: \[ e^{-y^2} \cdot \frac{dy}{dx} + \sin^2(x^2) \cdot 2x = 0 \].

Step 3: Solve for \( \frac{dy}{dx} \)

Rearrange the differentiated equation to isolate \( \frac{dy}{dx} \): \[ e^{-y^2} \frac{dy}{dx} = -2x \sin^2(x^2) \]. Then solve for \( \frac{dy}{dx} \) by dividing both sides by \( e^{-y^2} \): \[ \frac{dy}{dx} = -2x \sin^2(x^2) e^{y^2} \].

Step 4: Select the Correct Answer

Compare the solution \( \frac{dy}{dx} = -2x \sin^2(x^2) e^{y^2} \) to the provided options. The correct choice is (B) \(-2 x \sin^{2} x^{2} e^{y^{2}}\).

Key Concepts

Definite Integrals

Definite integrals are a way to calculate the accumulation of quantities, often representing areas under curves in the context of calculus. Unlike indefinite integrals, a definite integral has upper and lower limits. These limits, often represented by numbers, indicate the interval over which the function is being integrated. For example, in the equation \( \int_{0}^{y} e^{-t^{2}} dt \), the limits of integration are 0 and \( y \).

Understanding definite integrals is crucial because they translate the concept of finding an area under a curve into a precise value. When solving problems involving definite integrals:

• You identify the function and the interval of integration.
• Apply any necessary theorems, such as the Fundamental Theorem of Calculus, to evaluate the integral.
• Use integration rules and techniques to compute the result.

Implicit Differentiation

Implicit differentiation is a method used to find derivatives of functions that are not explicitly solved for one variable in terms of another. This technique is especially useful when dealing with equations where it is difficult or impossible to isolate one variable.

In the context of the given exercise, implicit differentiation was used to differentiate the equation \( \int_{0}^{y} e^{-t^{2}} dt + \int_{0}^{x^{2}} \sin^{2} t dt = 0 \) with respect to \( x \). By doing so, you treat \( y \) as a function of \( x \) and differentiate using chain rules:

• First, apply the chain rule to each integral individually.
• Differentiate any terms involving \( x \) directly.
• Use the relation \( \frac{dy}{dx} \) whenever you differentiate with respect to \( x \).

This approach helps to systematically find the derivative even when the function is not given in a straightforward manner.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is central to understanding the relationship between differentiation and integration in calculus. It serves as a bridge, connecting these two seemingly different concepts.

In the exercise, the theorem provides the basis for differentiating integrals:

• The FTC states that if \( f \) is continuous over \([a, b]\) and \( F \) is an antiderivative of \( f \), then \( \int_{a}^{b} f(t) dt = F(b) - F(a) \).
• When differentiating an integral with a variable limit, express it as \( f(u(x)) \cdot \frac{du}{dx} \).
• This approach allows us to convert the problem of differentiation into evaluating function values at the limits and simplifying accordingly.

For example, in our equation \( \int_{0}^{y} e^{-t^{2}} dt \), the FTC guides us to differentiate and find: \[ e^{-y^2} \cdot \frac{dy}{dx} \]. This reflects the power of the theorem in tackling problems that involve both derivatives and integrals.

Derivative Calculation

Calculating derivatives is one of the foundational operations in calculus. It involves finding the rate at which a function changes at any given point. In this exercise, calculating the derivative \( \frac{dy}{dx} \) is the ultimate goal.

To compute this, you use results from implicit differentiation and the FTC:

• After differentiating, rearrange the equation to isolate \( \frac{dy}{dx} \).
• Ensure that all terms involving \( x \) and \( y \) are correctly accounted for.
• Simplify the expression to match one of the given multiple-choice options.

Through accurate derivative calculation, you find that \( \frac{dy}{dx} = -2x \sin^2(x^2) e^{y^2} \), leading to the correct answer selection. Mastering derivative calculations ensures precision in determining function behavior, crucial in mathematics and applied sciences.