Problem 193
Question
Let \(\quad F(y)=\pi \int_{0} \sin (x y) d x\) Use Leibniz's rule to find \(\mathrm{F}^{\prime}(\mathrm{y})\).
Step-by-Step Solution
Verified Answer
Using Leibniz's rule, we find the derivative of \(F(y) = \pi \int_{0}^{\infty} \sin(xy) dx\) with respect to y as \(F'(y) = \pi \int_{0}^{\infty} x \cos(xy) dx\).
1Step 1: Identify the Limits of Integration and the Function to Differentiate
The limits of integration are 0 and \(\infty\), and the function inside the integral is \(\sin(xy)\).
2Step 2: Differentiate the Inside Function with Respect to y
Differentiate the function \(\sin(xy)\) with respect to y using the chain rule:
\(\frac{\partial}{\partial y} \sin(xy) = \cos(xy) \cdot \frac{\partial}{\partial y}(xy)\),
Now, differentiate the product xy with respect to y:
\(\frac{\partial}{\partial y}(xy) = x\).
So, the derivative of \(\sin(xy)\) with respect to y is:
\(\frac{\partial}{\partial y} \sin(xy) = x \cos(xy)\).
3Step 3: Apply Leibniz's Rule
Now, using Leibniz's rule, the derivative of F(y) with respect to y is:
\(F'(y) = \int_{0}^{\infty} \frac{\partial f(x, y)}{\partial y} dx = \int_{0}^{\infty} x \cos(xy) dx\).
4Step 4: Include the Constant Term
Finally, we must include the constant term \(\pi\) from the original integral. The final answer for F'(y) is:
\(F'(y) = \pi \int_{0}^{\infty} x \cos(xy) dx\).
We have now found the derivative of the integral using Leibniz's rule.
Key Concepts
Integration by PartsChain RuleLimits of IntegrationDifferentiation
Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based upon the product rule for differentiation and provides a way to transform the integral of a product of functions into other, hopefully simpler, integrals. The formula for integration by parts is given by:
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When choosing 'u' and 'dv', it is wise to select 'u' as the function that becomes simpler when differentiated, and 'dv' as the function that is easy to integrate. This method is particularly useful in solving the F(y) function from our exercise.
Chain Rule
The chain rule is fundamental in calculus for differentiating compositions of functions. It asserts that if you have a function such as h(x) = f(g(x)), then the derivative of h with respect to x is the derivative of f with respect to g(x), multiplied by the derivative of g with respect to x, symbolically:
\[\begin{equation}h'(x) = f'(g(x)) \times g'(x)\end{equation}\]
In our exercise, to find F'(y), we needed to differentiate \(\sin(xy)\) with respect to y. By applying the chain rule, we treated \(x\) as a constant and \(y\) as the variable function, leading us to the derivative \(x \cos(xy)\). Understanding this rule is essential for working with derivatives involving multiple functions and variables.
\[\begin{equation}h'(x) = f'(g(x)) \times g'(x)\end{equation}\]
In our exercise, to find F'(y), we needed to differentiate \(\sin(xy)\) with respect to y. By applying the chain rule, we treated \(x\) as a constant and \(y\) as the variable function, leading us to the derivative \(x \cos(xy)\). Understanding this rule is essential for working with derivatives involving multiple functions and variables.
Limits of Integration
Limits of integration specify the interval over which integration is performed. In the context of definite integration, they can greatly affect the value of the integral. It is also important to consider the behavior of the function at these limits, especially if either is infinite. In the exercise we are working on, the limits of integration are 0 to infinity (\(0\) to \(\infty\)), which means we are interested in the area under the function \(x \cos(xy)\) from x=0 to x approaching infinity. This requires students to have a strong understanding of the implications of infinity in calculus and proper techniques for evaluating such integrals, potentially using limits to find the behavior of the integral at infinity.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the function's rate of change. It is a core concept in calculus, used to analyze the way functions change and to solve a variety of real-world problems involving motion, growth, and approximation. In our example, we differentiated \(\sin(xy)\) with respect to y by applying the chain rule, which is one of the techniques in the toolkit of differentiation. Mastery of differentiation is crucial for students to progress in calculus and related fields, as it often forms the foundation for further study of more complex topics.
Other exercises in this chapter
Problem 191
a) Prove that if \(\mathrm{f}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then \(\mid \mathrm{f}(\mathrm{x})\) is Riemann integrable on
View solution Problem 192
Find the derivatives of a) \(\mathrm{x} \int_{1} \mathrm{t}^{2} \mathrm{dt} \quad\) with respect to \(\mathrm{x}\). b) \({ }^{\text {(t) } 2} \int_{1} \sin \lef
View solution Problem 195
Find \(\mathrm{F}^{\prime}(\mathrm{x})\) where a) \(F(x)=x \int_{0} e^{-(x) 2(t) 2} d t\) b) \(F(x)=\sin x \int_{(x) 2}\left(x^{2}-t^{2}\right)^{n} d t\)
View solution Problem 197
Proceed from the defintion of the Stieltjes integral to show that the function f given by $$ \begin{array}{cc} \mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})=0 &
View solution