Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. $$ \int x \cos x d x $$

Step-by-Step Solution

Answer

The result of \(\int x \cos x d x = x \sin x + \cos x + C\).

Step 1: Assign u and dv

Let \(u = x\), this makes \(dv = \cos x dx\).

Step 2: Compute du and v

Now, compute \(du = dx\) by differentiating \(u = x\) and also compute \(v = \int dv = \int \cos x dx = \sin x\) by integrating \(dv = \cos x dx\).

Step 3: Apply the integration by parts formula

Now use the integration by parts formula \(∫udv = uv - ∫vdu\). Substituting \(u, v, du\) we get \(\int x \cos x dx = x \sin x - \int \sin x dx\).

Step 4: Evaluating Integral

Evaluate the remaining integral to get \(\int x \cos x dx = x \sin x + \cos x + C\), where \(C\) is the constant of integration.

Key Concepts

Trigonometric Integrals

Trigonometric integrals are a vital part of calculus, involving the integration of functions composed of trigonometric functions such as sine, cosine, and tangent. These integrals often require special techniques to solve, as they are not always straightforward. In the context of the integral \( \int x \cos x \, dx \), the trigonometric function in question is \( \cos x \). When integrating expressions involving trigonometric functions, familiarity with their integral and derivative forms can be quite helpful. For example, the integral of \( \cos x \) is \( \sin x \), while its derivative is \( -\sin x \).

• Understand the basic antiderivatives for trigonometric functions.
• Recognize the need for integration techniques like substitution or parts for more complex integrals.

Knowing these relationships allows us to manipulate and simplify integrals during the solving process, especially when combined with polynomial functions, as seen in hybrid integrals involving products of polynomials and trigonometric functions.

Calculus Problem Solving

Solving calculus problems, particularly those involving integration, requires a structured approach to simplify and resolve complex expressions. Techniques such as integration by parts become essential for integrals that contain polynomial and trigonometric multiplicands.The formula for integration by parts is given by \( \int u \cdot dv = uv - \int v \cdot du \). This technique is especially useful in our example because we have a product of a polynomial \( x \) and a trigonometric function \( \cos x \). By identifying \( u = x \) and \( dv = \cos x \, dx \), we simplify the problem by following these steps:

• Differentiate \( u \) to find \( du \), and integrate \( dv \) to find \( v \).
• Apply the integration by parts formula to reduce the complexity of the integral.
• The process often simplifies the integral into a more manageable expression.

With practice, this problem-solving technique becomes an efficient tool for tackling various integrals beyond the basic cases, allowing deeper understanding and mastery of calculus concepts.

Definite and Indefinite Integrals

In calculus, integration can be classified into two main types: definite and indefinite integrals. The former involves the evaluation of the integral over specific bounds, providing a numerical value, while the latter expresses the general form of the antiderivative, including a constant of integration.For the integral \( \int x \cos x \, dx \), we specifically have an indefinite integral. This means we are seeking the antiderivative or the general formula that represents the original function. In solving indefinite integrals, it's crucial to include the constant of integration, \( C \), because indefinite integrals represent a family of solutions differing by a constant.

• Ensure you differentiate between definite and indefinite integrals—noting the requirement for bounds in definite integrals.
• Always add a constant \( C \) for indefinite integrals to represent all possible antiderivatives.

By understanding both types of integrals and their implications, you can confidently approach diverse calculus problems, applying the appropriate techniques for each scenario.