Problem 84 - Calculus and its applications

Question

Solve each integral. Each can be found using rules developed in this section, but some algebra may be required. $$ \int \sqrt[3]{64 x^{4}} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{12}{7} x^{7/3} + C \).

Step 1: Simplify the Integrand

Rewrite the integrand using fractional exponents. The expression \( \sqrt[3]{64 x^{4}} \) can be rewritten using fractional exponents as \( (64 x^{4})^{1/3} \).

Step 2: Apply Exponent Properties

Apply the properties of exponents to simplify \( (64 x^4)^{1/3} \). Break it down as \( 64^{1/3} \times (x^4)^{1/3} \), which is equal to \( 4 \times x^{4/3} \) because \( 64^{1/3} = 4 \) and \((x^4)^{1/3} = x^{4/3}\).

Step 3: Set Up the Simplified Integral

Using the simplified expression from the previous step, restate the integral: \( \int 4x^{4/3} \, dx \).

Step 4: Integrate Using Power Rule

Use the power rule to integrate. The power rule states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). For \( \int x^{4/3} \, dx \), apply the rule: \( \frac{x^{4/3 + 1}}{4/3 + 1} = \frac{x^{7/3}}{7/3} = \frac{3}{7}x^{7/3} \). Multiply by 4 to find the result: \( 4 \times \frac{3}{7}x^{7/3} = \frac{12}{7}x^{7/3} \).

Step 5: Integrate and Add Constant of Integration

Combine the results with the constant of integration \( C \). The final integral is \( \frac{12}{7} x^{7/3} + C \).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used to integrate expressions of the form \( x^n \). It's a straightforward method, and is vital for anyone learning integration. If you encounter an integral like \( \int x^n \, dx \), the Power Rule helps by providing this simple formula:

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

Here, \( n \) is any real number, and \( C \) represents the constant of integration, an essential part of indefinite integrals. This rule is applicable as long as \( n eq -1 \), because division by zero isn't possible.
When you use the Power Rule, you essentially increase the power of \( x \) by one, then divide the resulting term by this new power. It's like a two-step process that makes finding integrals remarkably simple and fast.
Understanding this rule fully helps you tackle more complex integrals, particularly when expressions have products or fractional exponents.

Fractional Exponents

Fractional exponents are another integral part of understanding calculus integration. They add flexibility in expressing and simplifying roots and powers.

• An expression like \( \sqrt[3]{64 x^{4}} \) can be rewritten as \( (64 x^{4})^{1/3} \), turning a root into a fractional exponent for easier manipulation.

Fractional exponents allow you to use all the familiar exponent rules but with added versatility.
For example, with \( (x^a)^b = x^{a \cdot b} \), having fractional exponents on bases allows operations like simplifying or distributing powers more smoothly.
Such transformations are crucial because they often lead to simpler forms that are easily integrated, especially when combined with other properties like the Power Rule. This makes them invaluable for calculus and algebraic simplification.

Algebra Simplification

Simplifying an algebraic expression is often the first and most crucial step in calculating integrals. It can involve rewriting expressions using simpler forms or reducing them into parts manageable by standard calculus rules.

• The original expression \( \sqrt[3]{64 x^4} \) was simplified to \( 4x^{4/3} \) by converting roots to fractional exponents and applying exponent rules to each term separately: \( 64^{1/3} = 4 \) and \( (x^4)^{1/3} = x^{4/3} \).

This process ensures your expressions are in the most manageable form for applying calculus techniques like the Power Rule.
A simplified integrand leads to a more straightforward integration process, minimizing errors and allowing you to focus on applying mathematical rules correctly.
Reducing complexity early on in calculations sets the stage for cleaner, more accurate results when working through indefinite or definite integrals.