Question

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{-\pi}^{\pi} x \sin x d x \text { (Express the answer in exact form.) } $$

Short answer

The integral of \(\int_{-\pi}^{\pi} x \sin x \, dx\) equals \(-2\pi\).

Step-by-step solution

Identify the Integral Type

The given problem is to compute the definite integral \( \int_{-\pi}^{\pi} x \sin x \, dx \). This is a definite integral of the product of \(x\) and \(\sin x\). We will need to use integration techniques suitable for these kinds of functions.

Choose the Integration Technique

For the integral \( \int x \sin x \, dx \), we use the technique of integration by parts. Integration by parts is given by the formula \( \int u \, dv = uv - \int v \, du \). Here, we choose \( u = x \) and \( dv = \sin x \, dx \).

Differentiate and Integrate

With \( u = x \), we differentiate to get \( du = dx \). For \( dv = \sin x \, dx \), we integrate to get \( v = -\cos x \).

Apply Integration by Parts

Using integration by parts, \( \int x \sin x \, dx = -x \cos x - \int (-\cos x) \, dx = -x \cos x + \int \cos x \, dx \).

Integrate Cosine Function

Integrate \( \int \cos x \, dx \) to get \( \sin x \). Thus, the integral becomes \( -x \cos x + \sin x + C \), where \( C \) is the constant of integration.

Evaluate the Definite Integral

Compute \( \left. \left( -x \cos x + \sin x \right) \right|_{-\pi}^{\pi} \). Calculate the values at the endpoints: \( \begin{align*}\text{At } x = \pi, & \; \pi (-1) + 0 = -\pi, \\text{At } x = -\pi, & \; -(-\pi)(-1) + 0 = \pi.\end{align*} \)

Compute the Result

Subtract the lower limit evaluation from the upper limit evaluation to find:\[(-\pi) - (\pi) = -2\pi.\]

Verification with Graphing Utility

Use a graphing calculator or software to verify if the integral from \(-\pi\) to \(\pi\) of \(x \sin x\) indeed calculates to \(-2\pi\).

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus, used to find the area under a curve between two specific points. In the exercise, we are asked to calculate the definite integral of the function \(x \sin x\) over the interval \([-\pi, \pi]\). This integral provides not just a number, but the notion of a 'net area' as it considers areas above the x-axis as positive and below it as negative.
To compute this integral, we utilize definite integration techniques, specifically integration by parts, due to the product of functions involved, \(x\) and \(\sin x\). The definite integral is evaluated by taking the anti-derivative of the function and then applying the Fundamental Theorem of Calculus, which comprises evaluating the resulting expression at the boundaries: \(x = \pi\) and \(x = -\pi\). The beauty of a definite integral is that the constant of integration vanishes when subtraction is performed directly between these evaluated endpoints.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially when dealing with integrals that involve sine, cosine, and other trigonometric ratios. In our specific problem, we focus on the function \(\sin x\). This function is known for its periodic nature, oscillating between -1 and 1 as x moves.
It plays a key role here, as our integral combines \(x\) with \(\sin x\). When employing integration by parts, we intentionally break down \(x \sin x\) into parts where \(u = x\) and \(dv = \sin x \, dx\). The derivative of \(x\) is straightforward, whereas integrating \(\sin x\) yields \(-\cos x\). These operations are crucial for applying the integration by parts method effectively. Understanding how trigonometric functions behave over different intervals aids in correctly evaluating the parts and simplifying the complex expression.

Graphing Utility Verification

Using graphing utilities like calculators or software allows students to visually confirm their algebraic results. After computing the integral of given functions, these tools graph the area under the curve, giving a visual affirmation of your work.
In our exercise, after arriving at the solution \(-2\pi\), we use a graphing utility to verify this result. By inputting the function \(x \sin x\) and setting the limits from \(-\pi\) to \(\pi\), one can easily visualize the graph and observe the calculated net area. The graph should reflect the symmetry about the y-axis due to the limits \(-\pi\) and \(\pi\). Such symmetry often helps in understanding why the areas above and below cancel each other out to result in \(-2\pi\). Graphing technology thus serves as an essential confirmatory tool in integrating functions and ensuring accuracy.