Question

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{\pi} x \cos x d x $$

Short answer

The definite integral evaluates to -2.

Step-by-step solution

Identify Integration Method

We start by identifying the integration method. The integral \( \int x \cos x \, dx \) suggests the use of integration by parts, as it involves a product of a polynomial \( x \) and a trigonometric function \( \cos x \).

Apply Integration by Parts Formula

The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). Choosing \( u = x \) (which simplifies upon differentiation) and \( dv = \cos x \, dx \) (easy to integrate), we find: \( du = dx \) and \( v = \sin x \).

Substitute into Integration by Parts Formula

Substituting into the formula, we get: \[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx. \]

Evaluate the Remaining Integral

Now evaluate the integral \( \int \sin x \, dx \), which is \(-\cos x + C \). Substitute this back:\[ \int x \cos x \, dx = x \sin x + \cos x + C.\]

Evaluate the Definite Integral

Use the expression derived and evaluate it from 0 to \( \pi \), i.e., \[ \int_{0}^{\pi} x \cos x \, dx = \left[ x \sin x + \cos x \right]_0^{\pi}. \]Plug in the limits:\( \pi \sin \pi + \cos \pi - (0 \cdot \sin 0 + \cos 0) = 0 - (-1) \).

Compute the Final Answer

Compute the final result:- At \( x = \pi \), the value is \( 0 + (-1) = -1 \).- At \( x = 0 \), the value is \( 0 - 1 = -1 \).Thus, the definite integral evaluates to \(-1 - (-1) = -2\).

Use a Graphing Utility for Confirmation

To confirm, use a graphing utility to compute the integral \( \int_{0}^{\pi} x \cos x \, dx \). The numerical result should match our calculated value, confirming its accuracy.

Key concepts

Integration by Parts

The integration by parts method is an essential tool in calculus, especially when dealing with integrals of products like that of a polynomial and a trigonometric function. It's akin to the product rule for differentiation.
Breaking it down, the formula is:- \( \int u \, dv = uv - \int v \, du \).First, we choose parts of the function for \( u \) and \( dv \):- **\( u \)**: generally selected for easy differentiation, e.g., polynomials like \( x \).- **\( dv \)**: typically a function easy to integrate, such as \( \cos x \).
In the exercise, selecting \( u = x \) and \( dv = \cos x \, dx \), we get:- \( du = dx \)- \( v = \sin x \) after integrating \( \cos x \).Substitute these into the integration by parts formula, solving step by step, reduces the main integral into simpler parts that can be integrated individually.

Polynomial and Trigonometric Functions

When dealing with definite integrals, particularly those involving polynomial and trigonometric functions, the challenge often lies in simplifying the function for integration.
In the given problem, we have a product of a polynomial, \( x \), and a trigonometric function, \( \cos x \). These functions are common in calculus problems. The polynomial multiplicands introduce differentiation ease, whereas trigonometric parts are integrable into known functions:

• \( \sin x \)
• \( \cos x \)
• \( \tan x \), etc.

Understanding these components helps streamline the integration process, paving the way for substitution and other methods like integration by parts. Balance between each function’s attributes leads to a smoother solution process and helps anticipate the kinds of transformations needed in complex integrals like the given one.

Graphing Utility

Graphing utilities, such as graphing calculators or software like Desmos, play a key role in confirming the computed results of definite integrals. These tools graph functions and compute integrals numerically.
This provides a visual confirmation of the calculated integral \( \int_{0}^{\pi} x \cos x \, dx \) from the original problem.
The process involves inputting the integral into the graphing utility, ensuring that the bounds from 0 to \( \pi \) are correctly set. The numerical result should match the analytical solution, \(-2\), providing confidence in the hand-computation:

• Draws the curve of the function shown to span over the specified limits.
• Shades the area representing the integral's value, visually confirming the evaluation.

By conducting this computational check, you can verify results and gain insights into the function's behavior across its domain, reinforcing understanding through visual context.