Problem 22 - Calculus: Early Transcendental Functions

Question

Evaluate the indicated integral. $$\int x^{2} \sec ^{2} x^{3} d x$$

Step-by-Step Solution

Answer

\(\frac{1}{3}tan(x^{3}) + C\)

Step 1: Identify the substitution

Let \(u = x^{3}\). Thus the derivative \(du/dx = 3x^{2}\) or, after rearranging, \(dx = du/(3x^{2})\).

Step 2: Substituting into the integral

Replace all instances of \(x^{3}\) and \(dx\) in the integral with their corresponding terms in 'u':\(\int x^{2} \sec^{2} u (du/3x^{2})\). The \(x^{2}\) terms cancel, leaving \(\int \sec^{2} u (du/3)\), which simplifies further to \(\frac{1}{3}\int \sec^{2} u du\).

Step 3: Calculate the integral in terms of 'u'

The integral of \(\sec^{2} u\) is \(tan(u)\), so the above expression becomes \(\frac{1}{3}tan(u) + C\) where 'C' is the constant of integration.

Step 4: Substitute 'u' back in terms of 'x'

Replace \(u\) with \(x^{3}\) in \(\frac{1}{3}tan(u) + C\), giving the final answer as \(\frac{1}{3}tan(x^{3}) + C\).

Key Concepts

Integral Calculus

Integral calculus focuses on finding the integral, or the antiderivative, of a function. It is the inverse process of differentiation. For this exercise, we aim to evaluate the integral \( \int x^{2} \sec ^{2} x^{3} \, dx \). Integrals can help us understand the area under curves, cumulative quantities, among other applications.
When solving integrals, one common method involves finding a function whose derivative gives us the original integrand. This approach is known as finding the antiderivative.

• Indefinite integrals, like in our exercise, result in a family of functions that include a constant of integration, \( C \), since the derivative of a constant is zero.
• In contrast, definite integrals have limits and yield a specific numeric value, often representing the area under a curve.

For our problem, we use integration by substitution—a technique we will discuss further—to find the indefinite integral, which simplifies the integrand by changing variables. Ultimately, this allows us to integrate a more straightforward function.

Trigonometric Integrals

Trigonometric integrals involve functions of trigonometric equations, in our case, the secant squared function, \( \sec^2 u \). These integrals often arise in problems involving angles and periodic phenomena. Understanding trigonometric identities and derivatives is crucial in evaluating them efficiently.
The antiderivative of \( \sec^2 u \) is \( \tan(u) \), a basic result from calculus.'' Knowing these derivatives and integrals allows us to solve problems more quickly and efficiently.

• Common trigonometric integrals include those of \( \sin(u) \), \( \cos(u) \), \( \tan(u) \), and \( \sec(u) \), each with specific antiderivatives.
• Mastery of these helps in tackling a variety of integration problems involving trigonometric functions.

In our exercise, once we expressed the integral in terms of \( u \), the integration simplified to a common trigonometric form, making the process much easier.

Techniques of Integration

Integration can often be complex, so developing robust techniques is vital. In this exercise, we used the technique of substitution. This method is beneficial when we identify part of the integrand as a function inside a differentiated component. By substituting \( u = x^3 \), we transformed the integral into one involving a simpler function.
After substitution, the differential \( dx \) is rewritten in terms of \( du \), simplifying the integration process substantially. Here are some steps typically involved in substitution:

• Identify the substitution: Choose a replacement that simplifies the integral.
• Rewrite the integrand and \( dx \) in terms of the new variable.
• Perform the integration in the new variable.
• Transform the result back in terms of the original variable.

During integration, other techniques such as integration by parts or partial fractions may come in handy, depending on the complexity of the integrand. Being equipped with a range of methods is key to tackling any integral challenge efficiently.