Problem 23 - Calculus and its applications

Question

Find each integral. $$ \int \frac{-7}{\sqrt[3]{x^{2}}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(-21x^{\frac{1}{3}} + C\).

Step 1: Rewrite the Integrand with Exponents

To simplify the integration, rewrite the integrand using exponential notation. The expression \( \frac{-7}{\sqrt[3]{x^{2}}} \) can be rewritten as \(-7x^{-\frac{2}{3}}\).

Step 2: Integrate Using the Power Rule

Use the power rule for integration, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Applying this to the integrand \(-7x^{-\frac{2}{3}}\), we have:\[\int -7x^{-\frac{2}{3}} \, dx = -7 \times \frac{x^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}\]

Step 3: Simplify the Integration Result

Calculate \(-\frac{2}{3}+1\), which is \( \frac{1}{3} \). Therefore, the integral becomes:\[-7 \cdot 3x^{\frac{1}{3}}\]Combining the constants, the result is:\[-21x^{\frac{1}{3}} + C\]

Step 4: Write the Final Answer

The final answer to the integration problem is:\[\int \frac{-7}{\sqrt[3]{x^{2}}} \, dx = -21x^{\frac{1}{3}} + C\]

Key Concepts

Power Rule for Integration

When dealing with integrals, the power rule for integration is a fundamental concept. It allows us to integrate expressions of the form \( x^n \) with ease. The rule states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \)

This formula is incredibly useful because it provides a straightforward method to find the antiderivative of polynomial expressions.
If you're dealing with fractional or negative exponents, don't worry! The power rule applies to these too, as long as the exponent \( n \) is not equal to \(-1\). When you apply the power rule, remember to adjust the exponent from \( n \) to \( n+1 \), and don't forget the ever-important constant of integration \( C \). This constant represents all possible vertical shifts of the antiderivative function derived from indefinite integrals.

Exponential Notation

Exponential notation is a powerful tool in simplifying complex expressions. It allows us to write repeated multiplication more compactly. For instance:

• \( x^3 \) is shorthand for \( x \times x \times x \)
• \( x^{-2} \) implies the reciprocal, or \( \frac{1}{x^2} \)

In integration, exponential notation helps to express roots and reciprocals in a form that is easier to integrate using rules like the power rule.
In the original problem, \( \sqrt[3]{x^2} \), which means the cube root of \( x^2 \), was converted to exponential form \( x^{\frac{2}{3}} \). This transformation simplifies the integration process, making it a routine application of the power rule for integration. Understanding how to convert between root notation to exponential notation can often make your calculus work much simpler.

Indefinite Integrals

Indefinite integrals are the reverse process of differentiation. While differentiation provides the rate of change of a function, integration helps us find the original function from its rate of change.

• The general form is \( \int f(x) \, dx \), which results in a family of functions due to the constant \( C \).

The result of an indefinite integral is not a single function but a family of functions. These are known as antiderivatives, represented by the addition of \( C \), which accounts for any possible vertical shifts.
In practice, this constant reflects the idea that there could be many functions with the same derivative. Indefinite integrals are essential in solving differential equations and analyzing the accumulated total of quantities that change continuously, such as area under a curve or total distance traveled over time.