91Ó°ÊÓ

Integration techniques are essential tools for solving integrals efficiently. Different methods help us tackle various forms of integrands. Some well-known techniques include:

• The power rule of integration
• Substitution
• Integration by parts
• Partial fractions
• Trigonometric substitutions

In this exercise, we use the power rule of integration due to the simplicity of the integrand after rewriting it in exponent notation. Remember, selecting the right technique can simplify the process and avoid unnecessary complications.

The power rule of integration is a fundamental technique for finding antiderivatives of expressions where the variable is raised to a power. The rule states that if you need to integrate a function of the form \( x^n \), where \( n \) is any real number and \( n \) is not equal to \(-1\), the integral can be found using the formula:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Here, \( C \) represents the constant of integration because the process of differentiation could have started from any constant value.

In our specific exercise, the integrand is \( \sqrt[3]{x} \), which we rewrite as \( x^{1/3} \. \) Using the power rule, we add 1 to the exponent \( \left(\frac{1}{3} + 1 = \frac{4}{3} \right)\). Then, we divide by the new exponent, yielding \[ \frac{x^{4/3}}{4/3} + C \. \]
Simplified further, it becomes \[ \frac{3}{4}x^{4/3} + C. \]

Exponent notation is a convenient way to express powers and roots. It helps simplify operations like integration. For example, \( x^a \) represents a number \( x \) raised to the power \( a \. \)

Roots can be expressed using fractional exponents:

• \( \sqrt{x} \) is \( x^{1/2} \)
• \( \sqrt[3]{x} \) is \( x^{1/3} \)
• \( \sqrt[n]{x} \) is \( x^{1/n} \)

By converting roots into fractional exponents, we can apply familiar rules of exponents to simplify expressions. In the given exercise, we converted \( \sqrt[3]{x} \) into \( x^{1/3} \) before using the power rule of integration.

Understanding exponent notation makes working with algebraic and calculus operations more manageable. It allows us to apply integration techniques methodically and avoid potential errors.