Problem 16 - Calculus: Early Transcendental Functions

Question

Evaluate the indicated integral. $$\int \frac{\cos (1 / x)}{x^{2}} d x$$

Step-by-Step Solution

Answer

The solution to the integral \( \int \frac{\cos (1 / x)}{x^{2}} dx \) is \( -\sin(1/x) \)

Step 1: Assigning Values for Substitution

Let's simplify the problem by substitution. Set \( u = 1/x \). Hence, \( du=-dx/x^2 \). Therefore, the integral becomes \( - \int \cos(u) du \)

Step 2: Evaluate Integral

This is a simple integral to evaluate. The integral of cos(u) is sin(u), so \( - \int \cos(u) du = -\sin(u) \)

Step 3: Substitute Back

Now replace u with \( 1/x \), so the final answer will yield : \( -\sin(1/x) \)

Key Concepts

Substitution Method

One of the most powerful techniques for solving integrals is the substitution method. It's a way to simplify an integral by changing variables, making it easier to integrate. When you encounter an integral that seems complex, like \( \int \frac{\cos (1 / x)}{x^{2}} \, dx \), substitution helps by transforming it into a simpler form. Here's a how-to:

• First, identify a part of the integral that can be substituted with a single variable, such as \( u = 1/x \). This simplifies expressions inside the integral.
• Next, calculate the differential \( du \). In this case, since \( u = 1/x \), differentiating gives \( du = -dx/x^2 \).
• Finally, substitute \( u \) and \( du \) into the integral. Our integral now looks like \( - \int \cos(u) \, du \). This is much easier to handle and can be integrated straightforwardly.

Substitution effectively rewrites the problem, allowing easier integration and providing a clear path to finding the solution of initially complicated integrals.

Trigonometric Integrals

Trigonometric integrals often involve integrals of sine, cosine, tangent, and other trigonometric functions. In calculus, these integrals can sometimes look daunting because they aren't linear. However, with the use of identities and substitution, they can be simplified.When integrating functions like \( \int \cos(u) \, du \), recall basic trigonometric integrals:

• The integral \( \int \cos(u) \, du \) is \( \sin(u) + C \), where \( C \) is the constant of integration.
• By knowing these basic forms, we can quickly solve integrals that would otherwise seem complex at first glance.

For our exercise, substituting transformed the original problem into a basic trigonometric integral \( -\int \cos(u) \, du = -\sin(u) \). This is a prime example of how trigonometric integrals, when combined with the substitution method, become manageable.

Definite and Indefinite Integrals

In calculus, you encounter both definite and indefinite integrals, each serving a distinct purpose.Definite integrals calculate the area under a curve between two points, along the x-axis. It is expressed as \( \int_a^b f(x) \, dx \), finding a number with limits of integration \( a \) and \( b \). Indefinite integrals, like the one in our exercise, are expressed as \( \int f(x) \, dx \). They represent a family of functions. The result includes an arbitrary constant \( C \) because integrating a derivative gives back a function plus a constant. For example, integrating \( \int \cos(u) \, du \) results in \( \sin(u) + C \).When solving the original indefinite integral \( \int \frac{\cos (1 / x)}{x^{2}} \, dx \), we used substitution to simplify and resulted in a function minus the integration constant since it wasn't required in the final substitution back, leading to \( -\sin(1/x) \). This highlights how an indefinite integral captures the essence of a derivative in reverse, providing the broader function landscape that would map to a given derivative structure.