Question

Prove that \(\int_{0}^{2 \lambda} \frac{\sin x}{x} d x=\int_{0}^{i} \frac{\sin 2 y}{y} d y=\frac{\sin ^{2} \lambda}{\lambda}+\int_{0}^{i} \frac{\sin ^{2} x}{x^{2}} d x .\) Deduce that \(\int_{0}^{\infty} \frac{\sin x}{x} d x=\int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x\) (It may be assumed that the integrals are convergent)

Short answer

Question: Prove the following relation between integrals: \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \int_{0}^{i} \frac{\sin 2y}{y}dy = \frac{\sin^{2}\lambda}{\lambda} + \int_{0}^{i} \frac{\sin^2 x}{x^2}dx\) And deduce that: \(\int_{0}^{\infty} \frac{\sin x}{x}dx = \int_{0}^{\infty} \frac{\sin^2 x}{x^2}dx\)

Step-by-step solution

STEP 1: Prove the first equality

In this step, we're trying to prove \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \int_{0}^{i} \frac{\sin 2y}{y}dy\). To begin, we need to apply a substitution method to the second integral. Let \(x = 2y\) and then \(dx = 2dy\). The integral becomes: \(\int_{0}^{i} \frac{\sin 2y}{y}dy = \frac{1}{2}\int_{0}^{2\lambda} \frac{\sin x}{x}dx\). After multiplying both sides of the equation by 2, we obtain the desired result: \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \int_{0}^{i} \frac{\sin 2y}{y}dy.\)

STEP 2: Prove the second equality

In this step, we need to prove \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \frac{\sin^2\lambda}{\lambda} + \int_{0}^{i} \frac{\sin^2 x}{x^2}dx\). To do this, recall the trigonometric identity \(\sin^2 x = \frac{1}{2}(1 - \cos2x)\). Now, we'll apply the fundamental theorem of calculus to differentiate the integral with respect to \(\lambda\): \(\frac{d}{d\lambda}\int_{0}^{2\lambda} \frac{\sin x}{x} dx = \frac{\sin 2\lambda}{2\lambda}\) We also differentiate the right side of the equation with respect to \(\lambda\): \(\frac{d}{d\lambda}(\frac{\sin^2\lambda}{\lambda} + \int_{0}^{i} \frac{\sin^2 x}{x^2}dx) = \frac{d}{d\lambda}(\frac{\sin^2\lambda}{\lambda})\) Using the trigonometric identity and the quotient rule, we get: \(\frac{d}{d\lambda}(\frac{\sin^2\lambda}{\lambda}) = \frac{2\lambda \sin\lambda \cos\lambda - \sin^2\lambda}{\lambda^2} = \frac{\sin 2\lambda}{2\lambda}\) Since the derivatives of both sides with respect to \(\lambda\) are equal, we can conclude that \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \frac{\sin^2\lambda}{\lambda} + \int_{0}^{i} \frac{\sin^2 x}{x^2}dx\).

STEP 3: Deduce the result for the improper integrals

Finally, we need to deduce that \(\int_{0}^{\infty} \frac{\sin x}{x}dx = \int_{0}^{\infty} \frac{\sin^2 x}{x^2}dx\). From Step 1 and Step 2, we have the following equality: \(\int_{0}^{2\lambda} \frac{\sin x}{x}dx = \frac{\sin^2\lambda}{\lambda} + \int_{0}^{i} \frac{\sin^2 x}{x^2}dx\) Now, let \(\lambda \longrightarrow \infty\). The term \(\frac{\sin^2 \lambda}{\lambda}\) converges to \(0\). Thus, \(\int_{0}^{\infty} \frac{\sin x}{x}dx = \int_{0}^{i} \frac{\sin^2 x}{x^2}dx\).

Key concepts

Integral Calculus

Integral calculus is a fundamental part of mathematical analysis that deals with the accumulation of quantities, such as areas under a curve, volumes, and other concepts that arise from summing infinitesimal elements. It is usually divided into two main types: definite and indefinite integrals. A definite integral has limits of integration and represents the accumulated quantity between those limits, while an indefinite integral, also known as an antiderivative, represents a family of functions with an arbitrary constant.

In the context of improper integrals, these are definite integrals where the interval of integration is unbounded, or the function being integrated has singularities on the interval. To evaluate an improper integral, we typically take limits where the unbounded behavior or singularity occurs, to determine if the integral converges (i.e., has a finite value) or diverges (i.e., increases without bound).

Sine Function

The sine function is a trigonometric function of an angle commonly used in various fields, including mathematics, physics, and engineering. It is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right-angled triangle. The sine function is periodic with a period of \(2\pi\) radians and oscillates between -1 and 1.

In integral calculus, the sine function often appears in integrals involving trigonometric expressions. The sine function's continuous and smooth behavior makes it particularly useful in modeling wave-like phenomena and periodic processes.

Convergence of Integrals

The concept of convergence of integrals is crucial when dealing with improper integrals. An integral converges if the limit of the integral, as one or both bounds approach infinity or a point of discontinuity, exists and is finite. Conversely, an integral diverges if this limit does not exist or is infinite.

To determine the convergence of integrals involving trigonometric functions like sine, we may need to analyze the behavior of the function as the variable approaches infinity or a point of discontinuity, and apply specific theorems or criteria for convergence such as the comparison test or the p-test.

Substitution Method in Integration

The substitution method, also known as u-substitution, is a technique used to simplify the process of integration. This method involves changing the variable of integration to a new variable that transforms the integral into an easier one to evaluate. In effect, it is the reverse process of the chain rule in differentiation.

For trigonometric integrals, substitution can be particularly effective when a function is composed with a trigonometric function. By substituting a part of the integral with a new variable, you can often reduce the complexity of the integrand and turn a challenging integral into a more manageable form.

Trigonometric Identities

Trigonometric identities are equalities that involve trigonometric functions and are true for all values of the variables for which both sides of the equality are defined. These identities are invaluable tools in simplifying and evaluating expressions and integrals involving trigonometric functions.

Some common trigonometric identities include Pythagorean identities, angle sum and difference formulas, double angle formulas, and half-angle formulas. For example, the identity \(\sin^2 x = \frac{1}{2}(1 - \cos(2x))\) is a double angle identity that is used to express \(\sin^2 x\) in terms of \(\cos(2x)\), which can be particularly useful in integrating expressions involving \(\sin^2 x\).