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Chapter 1: Problem 7

Now evaluate the following integrals. \(\int \frac{\cos \left(\frac{3}{x}\right)}{x^{2}} d x\)

Short Answer

Expert verified
-\(\sin \left(\frac{3}{x}\right) + C\)

Step by step solution

01

Identify function for u-substitution

Let \(u\) be the function inside the cosine function, i.e \(u=\frac{3}{x}\), and consequently, \(du = -\frac{3}{x^{2}} dx\).
02

Rewrite the integral in terms of \(u\)

After multiplying both sides of the last equation of Step 1 by -1, it implies \(du = - \frac{3}{x^{2}} dx\). This is considering that the integral of \(dx\) can be rewritten in terms of \(u\), changing the integral to \(-\int \cos(u) du\).
03

Evaluate the integral

The antiderivative of \(\cos(u)\) is \(\sin (u)\). Because of the minus sign the antiderivative is -\(\sin (u) + C\) where \(C\) is the constant of integration.
04

Back-substitute

We then replace \(u\) with \(\frac{3}{x}\) yielding -\(\sin \left(\frac{3}{x}\right) + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding u-substitution
U-substitution is a valuable technique in calculus used to simplify the process of integration. It involves substituting a part of the integral with a new variable, usually denoted as \(u\). This technique works well when you have a function inside another function, like our example where \(u = \frac{3}{x}\).
  • You choose \(u\) to make the integral easier.
  • Find the derivative \(du\) and express it in terms of \(dx\).
  • Rewrite the entire integral in terms of \(u\) and \(du\).
This transformation allows us to focus on a simpler integral form, making the calculation more manageable.
Exploring Antiderivatives
The antiderivative is the opposite of differentiation; it involves finding a function whose derivative is the given function. In our exercise, we need the antiderivative of \(\cos(u)\), which is \(\sin(u)\). The process looks like this:
  • Identify the integrand (function to integrate).
  • Find a function whose derivative is your integrand.
By understanding the basic antiderivatives of common functions, like trigonometric functions, you can solve integrals more efficiently.
Working with Trigonometric Integrals
Trigonometric integrals involve integrands that include trigonometric functions like sine, cosine, tangent, etc. These integrals often require specific techniques like u-substitution for simplification. In this exercise, we work with \(\cos(\frac{3}{x})\). After substitution, the integral becomes \(-\int \cos(u) du\).
  • Recognize the need for trigonometric identities or substitutions.
  • Simplify using known antiderivatives, such as \(\int \cos(u) \), which equals \(\sin(u)\).
Approaching trigonometric integrals with these strategies can significantly streamline the integration process.
The Role of the Constant of Integration
When we find antiderivatives, we include a constant of integration \(C\) in our final answer. This constant represents the indefinite nature of integration, as many functions have the same derivative.
  • Recognize that integration could yield a family of curves.
  • Always add \(C\) to indicate general solutions.
In our solution, after integrating \(-\cos(u)\), we add \(C\) to get \(-\sin(u) + C\), ensuring we cover all possible solutions.

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