Question

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt[3]{x}-\frac{1}{2 \sqrt[3]{x}}\right) d x $$

Short answer

The indefinite integral of \( \sqrt[3]{x} - \frac{1}{2\sqrt[3]{x}} \) is \( F(x) = \frac{3}{4}x^{4/3} - 3x^{2/3} + C \)

Step-by-step solution

Identify and Write Down the Integral Terms

Separate the integral and write down the integral of each term separately. So, \( \int\left(\sqrt[3]{x}\right) d x \) and \( \int\left(-\frac{1}{2 \sqrt[3]{x}}\right) d x \).

Calculate the Integrals

The integral of \( \sqrt[3]{x} \) is \( \frac{3}{4}x^{4/3} + C_1 \) and the integral of \( -\frac{1}{2\sqrt[3]{x}} \) is \( -3x^{2/3} + C_2 \).

Combine the Results

Sum up the results from step 2 to get the final answer \( F(x) = \frac{3}{4}x^{4/3} - 3x^{2/3} + C \). Here, \( C \) is the sum of the constant of integration from each integral term, i.e. \( C = C_1 + C_2\).

Differentiate to Check

Differentiate the result \( F(x) \) using the power rule. This results in \( F'(x) = \sqrt[3]{x} - \frac{1}{2\sqrt[3]{x}} \), which is indeed the original integrand, confirming that the answer is correct.