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Chapter 11: Problem 4

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(1-\frac{1}{\sqrt[3]{x^{2}}}\right) d x=x-3 \sqrt[3]{x}+C $$

Short Answer

Expert verified
The given statement has been verified. The derivative of the right side is equal to the integrand of the left side, i.e., \(1-\frac{1}{\sqrt[3]{x^{2}}}\).

Step by step solution

01

Identify the right side of the given statement

The right side of the given statement is \(x-3\sqrt[3]{x}+C\). This is the function that must be differentiated.
02

Differentiate the right side

The derivative of \(x\) is \(1\), the derivative of \(-3\sqrt[3]{x}\) is \(-\frac{1}{\sqrt[3]{x^{2}}}\), and the derivative of any constant is \(0\). Therefore, the derivative of the right side is \(1-\frac{1}{\sqrt[3]{x^{2}}}\).
03

Compare the derivative with the integrand

The derivative obtained in step 2 is \(1-\frac{1}{\sqrt[3]{x^{2}}}\), which is indeed the integrand provided on the left side of the statement. This means that the derivative of the right side is equal to the integrand of the left side, verifying the given statement.

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