Problem 111 - University Calculus: Early Transcendentals

Question

Suppose that \(F(x)\) is an antiderivative of \(f(x)=(\sin x) / x\) \(x>0 .\) Express. $$\int_{1}^{3} \frac{\sin 2 x}{x} d x$$ in terms of \(F\)

Step-by-Step Solution

Answer

\( \frac{1}{2} (F(6) - F(2)) \)

Step 1: Recognize the Inner Function

Identify the integral's function, which is \( \int \frac{ \sin 2x}{x} \, dx \). Notice that \( \frac{ \sin 2x}{x} \) resembles \( f(x) = \frac{\sin x}{x} \) with an argument substitution of \( 2x \).

Step 2: Perform Substitution

We substitute \( u = 2x \), which implies \( du = 2 \, dx \) or \( dx = \frac{1}{2} du \). This changes the integral bounds accordingly: when \( x = 1 \), \( u = 2 \cdot 1 = 2 \) and when \( x = 3 \), \( u = 2 \cdot 3 = 6 \).

Step 3: Rewrite the Integral with Substitution

Using the substitution in Step 2, rewrite \( \int_{1}^{3} \frac{\sin 2x}{x} \, dx \) as \( \int_{2}^{6} \frac{\sin u}{u} \cdot \frac{1}{2} \, du \). Factor out the constant \( \frac{1}{2} \); this simplifies the integral to \( \frac{1}{2} \int_{2}^{6} \frac{\sin u}{u} \, du \).

Step 4: Express the Integral in Terms of F

Since \( F(x) \) is an antiderivative of \( f(x) = \frac{\sin x}{x} \), the integral \( \int_{2}^{6} \frac{\sin u}{u} \, du \) can be expressed as \( F(6) - F(2) \). Substitute this into the expression: \( \frac{1}{2} (F(6) - F(2)) \).

Step 5: Final Expression

Combine all the previous steps to find that \( \int_{1}^{3} \frac{\sin 2x}{x} \, dx = \frac{1}{2} (F(6) - F(2)) \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a valuable technique used in calculus to simplify the integration of certain functions involving trigonometric expressions. In the context of the given exercise, it plays a crucial role by transforming the integral into a more manageable form. Here, we notice the similarity between the original function, \( \frac{\sin 2x}{x} \), and \( \frac{\sin x}{x} \) by considering a substitution based on the argument change.

• The function \( \sin 2x \) can be expressed in terms of \( \sin u \) by letting \( u = 2x \), which transforms the trigonometric identity into a simpler form for integration.
• This substitution adjusts the variable of integration, \( x \), to \( u \), which helps align the structure of the integral with the known form \( f(x) = \frac{\sin x}{x} \).

The substitution is made possible due to the trigonometric property where doubling the angle multiplies the variable inside the \( \sin \) function. This insight allows us to perform integration using substitution with a trigonometric adjustment, effectively splitting the problem into more straightforward parts.

Antiderivative

An antiderivative is a fundamental concept in calculus, representing a function whose derivative is the original function. In this exercise, we are given that \( F(x) \) is an antiderivative of the function \( f(x) = \frac{\sin x}{x} \).

• Understanding that an antiderivative reverses the process of differentiation is key. It helps us recognize that integration is seeking out this reverse operation.
• The function \( F(x) \) will provide the value of the integral over any interval, by evaluating \( F(b) - F(a) \) when integrating from \( a \) to \( b \).

For this specific problem, finding an antiderivative means expressing the result of a definite integral, that originally looked complex, in terms of the simpler form \( F(x) \). The antiderivative makes the computation of the bounded area much more straightforward by simplifying the evaluation without directly solving the integral from scratch each time.

Change of Variables

The change of variables, also known as substitution, is a technique used to transform a complex integral into a simpler one. It is especially useful when dealing with fractions or products involving trigonometric functions.Consider the integral \( \int \frac{\sin 2x}{x} \, dx \). By substituting \( u = 2x \), the differential changes as well, allowing \( du = 2 \, dx \), thus \( dx = \frac{1}{2} \, du \). This substitution modifies the bounds:

• When \( x = 1 \), \( u = 2 \cdot 1 = 2 \).
• When \( x = 3 \), \( u = 2 \cdot 3 = 6 \).

With these substitutions, the integral can then be re-written as \( \int_{2}^{6} \frac{\sin u}{u} \cdot \frac{1}{2} \, du \), simplifying the problem. This technique systematically reduces the complexity of the integral by realigning the problem to match a known antiderivative, thus allowing the use of simpler, existing solutions.