Problem 59 - Essential Calculus

Question

State the integration formula you would use to perform the integration. Do not integrate. $$ \int \sqrt[3]{x} d x $$

Step-by-Step Solution

Answer

The formula that will be used is: \( \int x^n dx = \frac{1}{n+1} x^{n+1} + C \), substituting \( n = 1/3 \) to accommodate the function \( f(x) = \sqrt[3]{x} \) or \( f(x) = x^{1/3} \).

Step 1: Identify function to integrate

Here we are asked to find the formula for the integral of the function \( f(x) = \sqrt[3]{x} \). In order to state the correct integration formula, we need first to recognize this function within the existing integration formulas.

Step 2: Recognize function as a power function

The function \( f(x) = \sqrt[3]{x} \) can be rewritten as a power function as \( f(x) = x^{1/3} \). This is useful because one common category of integrable functions includes power functions, written in the form \( f(x) = x^n \), and there are specific rules to integrate such functions.

Step 3: State the power function integration formula

The integration formula for power functions with \( n \neq -1 \) is given by: \( \int x^n dx = \frac{1}{n+1} x^{n+1} + C \), where \( C \) is the constant of integration and is always included in indefinite integrals. This formula would be used to actually perform the integration of \( f(x) \).