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Chapter 11: Problem 31

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

Expert verified
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

01

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 \( \int\left(-\frac{1}{2 \sqrt[3]{x}}\right) d x \).
02

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 \( -3x^{2/3} + C_2 \).
03

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. \( C = C_1 + C_2\).
04

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.

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Most popular questions from this chapter

Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}\) (a) 1 (b) 6 (c) \(-3\) (d) 3 (e) 4

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