Problem 9 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int \frac{x^{1 / 3}-3}{x^{2 / 3}} d x$$

Step-by-Step Solution

Answer

The general antiderivative of the given expression is \(\frac{1}{2}x^2 - 9x^{1/3} + C\).

Step 1 - Split the expression

In this step, the expression \(\int \frac{x^{1 / 3}-3}{x^{2 / 3}} d x\) is split into two simpler fractions. It turns out to be \(\int \frac{x^{1/3}}{x^{2/3}} dx - \int \frac{3}{x^{2/3}} dx\).

Step 2 - Simplify the fractions

The next step is to simplify the fractions. \(\frac{x^{1/3}}{x^{2/3}}\) simplifies to \(x\), and \(\frac{3}{x^{2/3}}\) simplifies to \(3x^{-2/3}\). Hence, the expression becomes \(\int x dx - \int 3x^{-2/3} dx\).

Step 3 - Find the antiderivatives

To find the antiderivative, integrate each term. The integral of \(x\) is \(\frac{1}{2}x^2 + C\), and the integral of \(3x^{-2/3}\) is \(-9x^{1/3} + C\).

Step 4 - Combine the antiderivatives

By combining the results of the integrations of the two terms, the general antiderivative of the original expression is \(\frac{1}{2}x^2 - 9x^{1/3} + C\)

Key Concepts

Integration

Integration is a fundamental concept in calculus that establishes the relation between a function and its antiderivative. The process of finding an antiderivative is often referred to as 'finding the indefinite integral.' The whole purpose of integration is to essentially perform the reverse operation of differentiation. When we integrate, we are accumulating the quantities that, when differentiated, would yield the function we started with. In our exercise, the integration of the function \(\frac{x^{1 / 3}-3}{x^{2 / 3}} dx\) entails reversing the process to find the function whose derivative gives us the initial expression.

The importance of integration is not limited to just solving math problems. It has pivotal applications in fields like physics, where it is used to calculate quantities like areas under curves, volumes, and in problems involving rates of change, such as velocity and acceleration.

Algebraic Simplification

Algebraic simplification is a useful technique in integration, as it can make complex expressions more manageable before the integration process begins. Simplification often involves applying basic algebraic rules to transform expressions, such as combining like terms, factoring, and canceling common factors. It is crucial to simplify expressions properly to avoid errors later in the integration process. In our example, the fraction in the integrand, \(\frac{x^{1 / 3}-3}{x^{2 / 3}}\), is simplified by splitting it into two separate terms and then reducing each term to its simplest form. This was achieved by recognizing that exponent rules allow us to divide powers of the same base by subtracting exponents, thus turning the complex fraction into two simpler terms for easier integration.

Integral of Power Functions

The integral of power functions involves integrating functions of the form \(x^n\), where \(n\) is any real number. For most values of \(n\), except when \(n = -1\), the antiderivative can be found using the power rule for integration. This rule states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is an arbitrary constant. In the context of our textbook problem, we are dealing with the power functions \(x^{1/3}\) and \(x^{-2/3}\). Applying the power rule, we can readily find their antiderivatives. The power rule simplifies the integration process and can be widely applied, making it an invaluable tool in calculus.

Constants of Integration

The constant of integration, denoted by \(C\), represents an arbitrary constant that arises when computing an indefinite integral. Since the derivative of a constant is zero, every time we integrate, there is potentially an unknown additive constant we must include. This constant reflects the fact that an infinite number of antiderivatives exist for a given function, differing only by a constant term.

In concluding the integration process for our exercise, we need to combine the constants from integrating each term into a single constant. This is because, in the indefinite integral, the constant may represent any number, and thus it isn't necessary to keep multiple terms. The final result of our antiderivative includes this combined constant of integration denoted simply as \(C\) in the expression \(\frac{1}{2}x^2 - 9x^{1/3} + C\). The correct understanding of constants of integration is critical not only in finding antiderivatives but also in solving initial value problems where the constant is determined by specific conditions.