Problem 16 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

\( \int \sqrt[3]{x^2} \, dx = \frac{3}{5} x^{5/3} + C \)

Step 1: Identify the Integral Form

The integral given is \( \int \sqrt[3]{x^2} \, dx \). Recognize that \( \sqrt[3]{x^2} \) can be rewritten using exponent notation as \( x^{2/3} \).

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \). Substitute \( n = \frac{2}{3} \) into this formula.

Step 3: Compute the Integral

Using the power rule, the integral becomes \( \int x^{2/3} \, dx = \frac{x^{(2/3)+1}}{(2/3)+1} + C \).

Step 4: Simplify the Expression

Calculate \( 2/3 + 1 = 5/3 \). Therefore, the integral becomes \( \frac{x^{5/3}}{5/3} + C \). This can be further simplified by multiplying the numerator and denominator by \(3\): \( \frac{3}{5} x^{5/3} + C \).

Key Concepts

Power Rule

The Power Rule is a straightforward and essential technique in calculus, widely used for finding integrals of expressions of the form \( x^n \). The power rule for integration is different from the differentiation power rule. Here, when you integrate \( x^n \), the formula to apply is:

• \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Where:

• \( n \) is any real number except \( n = -1 \) (because this would lead to division by zero).
• \( C \) is the constant of integration, essential for indefinite integrals.

When using the power rule, it's important to adjust the exponent correctly and ensure that the increment is applied to both the numerator and denominator. This rule simplifies the process of working with polynomial expressions and expressions that can be rewritten as polynomial-like through exponent notation.

Exponent Notation

Exponent notation is a way of expressing numbers and expressions using powers or exponents. It's a useful tool that helps simplify complex expressions, making them easier to manage within mathematical operations, including integration. In this exercise:

• We have \( \sqrt[3]{x^2} \), which can be rewritten as \( x^{2/3} \) using exponent notation.

By converting roots and radicals into fractional exponents, you can directly apply the power rule for integration. The critical aspect here is understanding that the denominator of the fractional exponent reflects the root, while the numerator shows the power of the variable. This conversion allows you to take advantage of the established integration techniques for easier computation and greater accuracy in solving integral problems.

Integral Simplification

Simplifying an integral is essential for interpreting the solution and ensuring that it's in the most understandable form. After applying the power rule to the expression \( x^{2/3} \), we obtained:

• \[ \int x^{2/3} \, dx = \frac{x^{5/3}}{5/3} + C \]

Simplification involves cleaning up this expression by eliminating complex fractions. Multiply both the numerator and the denominator by the reciprocal of the denominator's term (in this case 3) to simplify:

• \[ \frac{3}{5} x^{5/3} + C \]

This final expression is more straightforward, avoids fractional exponents in the denominator, and is easier to interpret. Simplification ensures the integral solution is clean and ready for applications or further mathematical manipulation.