Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. $$ \int x \sec ^{2} x d x $$

Step-by-Step Solution

Answer

\The integral of \( x \sec^{2}(x) dx \) is \( x \tan(x) + \ln |\cos(x)| + C \) where \( C \) is the constant of integration.

Step 1: Identify \( u \) and \( dv \)

Here, by using the method of integration by parts, the function \( x \sec^{2}(x) \) can be separated into two parts as \( u \) and \( dv \). So, choose \( u = x \) and \( dv = \sec^{2}(x) dx \) for this particular problem.

Step 2: Find \( du \) and \( v \)

The differential of \( u \) or \( du \) is the derivative of \( x \) with respect to \( x \), which is \( dx \). For \( v \), you need to know that the integral of \( \sec^{2}(x) \) is \( \tan(x) \). So, \( v = \tan(x) \).

Step 3: Apply the formula of integration by parts

Substitute \( u \), \( dv \), \( du \), and \( v \) into the integration by parts formula: \(\int u dv = uv - \int v du\). This results in \( x \tan(x) - \int \tan(x) dx \).

Step 4: Evaluate the integral

Now it's time to evaluate the remaining integral, which is \(\int \tan(x) dx \). The integral of \( \tan(x) \) is \( -\ln |\cos(x)| \).

Step 5: Summarize the result

Substitute this result back into the original equation. The final answer when the term \(-\ln |\cos(x)|\) is added is \( x \tan(x) -(-\ln |\cos(x)|) = x \tan(x) + \ln |\cos(x)| + C\), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, tangent, and their variations. Understanding these functions is crucial for evaluating integrals where trigonometric expressions are present. In our exercise, we specifically look at the integral of \(x \sec^2(x)\). The task is to use known trigonometric identities and integration techniques to solve this.

- **Key Trigonometric Functions:** Sine \(\sin(x)\), cosine \(\cos(x)\), tangent \(\tan(x)\), and secant \(\sec(x)\) are the most common trigonometric functions.- **Secant and Tangent Relationship:** Remember that \(\sec^2(x)\) is the derivative of \(\tan(x)\), which simplifies integration tasks. What's critical here is recognizing this relationship when performing integrals involving trigonometric functions.

In our solution, recognizing that the derivative of \( \tan(x) \) is \( \sec^2(x) \) allows us to progress with integration by parts, making the problem simpler.

Calculus Problem Solving

Calculus problem solving often involves breaking down a complex function into more manageable parts. That's where integration by parts becomes a valuable tool. In our given exercise, the integral \(\int x \sec^2(x) \, dx \) is simplified using this method.

Here's the approach we took:

• **Identify Components:** We select \( u = x \) and \( dv = \sec^2(x) \, dx \). This division makes it easier to apply the formula \( \int u \, dv = uv - \int v \, du \).
• **Differentiate and Integrate:** By differentiating \( u \) to get \( du \) and integrating \( dv \) to get \( v \), we establish the components of the integration by parts formula.
• **Apply the formula:** Substitute back into the formula and simplify, ultimately reducing the integral to a simpler form.

Integration by parts transforms a tough integral into one that is much simpler, utilizing the derivative and antiderivative relationships. This method demonstrates the systematic breakdown of calculus problem-solving techniques.

Integral Calculus

Integral calculus focuses on the concept of integration and its applications. It's the branch of calculus that deals with finding the area under a curve or solving differential equations using integrals.

In the provided exercise, we delve into:

• **Integration Techniques:** Students often utilize methods like substitution or integration by parts to tackle more difficult integrals. The method chosen depends on the form of the function.
• **Finding Antiderivatives:** In straightforward terms, solving an integral means finding its antiderivative. This provides a function whose derivative matches the integrand.
• **Constant of Integration:** The solution involves adding a constant \( C \), which reflects the general solution of an indefinite integral. This indicates that there are infinitely many antiderivatives differing by a constant.

By using integration by parts, we essentially decomposed the task of integrating \(x \sec^2(x)\) into simpler operations, culminating in a manageable expression that includes the logarithm of the cosine function. This highlights the essence of integral calculus: breaking down complex integrations into solvable segments while understanding the overall behavior of functions.