Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int \sqrt[3]{x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \sqrt[3]{x^{2}} \) is \( \frac{3}{5}x^{5/3} + C \). Checking the result by differentiating, we obtained the original function \( \sqrt[3]{x^{2}} \), implying that the integral was calculated correctly.

Step 1: Identify the function to be integrated

The function given that needs to be integrated is \( \sqrt[3]{x^{2}} \)

Step 2: Convert the root to an exponent

It is more simple to handle an exponent rather than a root. The fraction 2/3 will be used in place of the cubic root, resulting in \(x^{2/3}\)

Step 3: Apply integral rules

According to the power rule for integration, which states that the integral of \(x^{n}\) is \(x^{n+1}/(n+1)\), apply this rule to \(x^{2/3}\) to get the integral.

Step 4: Compute the Integral

Integrating \(x^{2/3}\), you obtain \( \frac{3}{5}x^{5/3} + C\), where C is the constant of integration.

Step 5: Differentiate to Check

Next, differentiate the result \(\frac{3}{5}x^{5/3} + C\) back to check if it gives the integrand \(x^{2/3}\). According to the power rule of differentiation, the derivative \( \frac{5}{3}*\frac{3}{5}x^{(5/3-1)} \) becomes \(x^{2/3}\). This confirms that the calculation of the integral is correct.