Question

Given that \(\int_{0}^{\infty} \frac{d x}{y^{2}+x^{2}}=\frac{\pi}{2 y},\) differentiate with respect to \(y\) and so evaluate \(\int_{0}^{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}}.\)

Short answer

\[ \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} = \frac{\pi}{4 y^{3}}.\]

Step-by-step solution

Given Integral

Start with the given integral \[ I(y) = \int_{0}^{\d}{\infty} \frac{d x}{y^{2}+x^{2}}=\frac{\pi}{2 y} \]

Differentiate Both Sides

To find the integral involving \frac{1}{\left(y^{2}+x^{2}\right)^{2}}, differentiate both sides of the given equation with respect to \(y\).

Differentiate the Left Side

Differentiate the left side of the equation under the integral sign using Leibniz's rule: \[ \frac{d}{d y} \int_{0}^{\d}{\infty} \frac{d x}{y^{2}+x^{2}} = \int_{0}^{\d}{\infty} \frac{\partial}{\partial y}\left( \frac{1}{y^{2}+x^{2}} \right)dx ewline = \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} \left(-2y\right) ewline = - 2 y \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}}.\]

Differentiate the Right Side

Differentiate the right side of the given equation to find: \[ \frac{d}{d y} \left( \frac{\pi}{2 y} \right) = \frac{d}{d y} \left( \frac{\pi}{2} \frac{1}{y} \right) = - \frac{\pi}{2 y^{2}}.\]

Set Both Sides Equal

Set both differentiated sides equal to each other: \[ - 2 y \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} = - \frac{\pi}{2 y^{2}}.\]

Solve for Integral

Solve for \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} by isolating it: \[ \int_{0}^{\d}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} = \frac{\pi}{4 y^{3}}.\]

Key concepts

Leibniz's rule

Leibniz's rule is a mathematical method used for differentiating an integral whose limits are constants. It allows us to take the derivative with respect to a parameter inside the integral. This is particularly useful when dealing with integrals that depend on an external parameter.
In this exercise, we apply Leibniz's rule to differentiate the integral \ \ I(y) = \int_{0}^{\infty} \frac{d x}{y^{2}+x^{2}} \ with respect to \(y\).
This means we need to find \(\frac{d}{d y} \int_{0}^{\infty} \frac{d x}{y^{2}+x^{2}}\).
Using Leibniz's rule, we can rewrite the derivative inside the integral:
\( \frac{d}{d y} \int_{0}^{\infty} \frac{d x}{y^{2}+x^{2}} = \int_{0}^{\infty} \frac{\partial}{\partial y}\left( \frac{1}{y^{2}+x^{2}} \right)dx \). This reformation simplifies our calculations and helps deeply understand the integral's dependency on parameter \(y\).

Differentiation under the integral sign

Differentiation under the integral sign is a technique where we interchange the operations of differentiation and integration. In simpler terms, we compute the derivative of an integral by initially inserting it inside the integral.
In this problem, we start with the integral:
\( \int_{0}^{\infty} \frac{d x}{y^{2}+x^{2}} = \frac{\pi}{2 y} \).
To find the related integral \( \int_{0}^{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} \), we differentiate both sides with respect to \(y\).
Inside the integral, this becomes:
\( \frac{\partial}{\partial y}\left( \frac{1}{y^{2}+x^{2}} \right) = \left( \frac{-2y}{\left(y^{2}+x^{2}\right)^{2}} \right)\). This enables us to tackle more complex integrals by breaking them down into simpler differentiation problems.

Definite integrals

A definite integral has specific upper and lower limits, and it represents the area under a curve between these points. In the context of our exercise, we have the integral:
\( \int_{0}^{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)} \).
This specific integral depends on the parameter \(y\) and spans from 0 to infinity.
In the solution, differentiating the integral under the sign gives us:
\( \int_{0}^{\infty} \frac{-2y}{\left(y^{2}+x^{2}\right)^{2}} d x \).
We can handle these definite integrals elegantly using Leibniz's rule and partial differentiation techniques, as evaluating the integral directly might be challenging.

Partial differentiation

Partial differentiation refers to finding the derivative of a function with respect to one variable while keeping other variables constant. It serves as a cornerstone in handling functions of multiple variables.
In this exercise:
\( \frac{\partial}{\partial y}\left( \frac{1}{y^{2}+x^{2}} \right) \).
We treat \(x\) as a constant and differentiate with respect to \(y\). The resulting partial derivative is:
\( \frac{-2y}{\left(y^{2}+x^{2}\right)^{2}} \).
This partial differentiation steers the step-by-step integration under the integral sign technique.
Later it empowers us to isolate and solve the integral easily:
\( \int_{0}^{5}{\infty} \frac{d x}{\left(y^{2}+x^{2}\right)^{2}} = \frac{\pi}{4 y^{3}} \). This highlights how partial differentiation plays a pivotal role in manipulating and solving integrals with dependent parameters.