Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(4 x^{3}-\frac{1}{x^{2}}\right) d x=x^{4}+\frac{1}{x}+C $$

Step-by-Step Solution

Answer

Yes, the statement is correct. The derivative of the right side of the equation \(4x^{3}-\frac{1}{x^{2}}\) is identical to the integrand of the left side.

Step 1: Evaluate the Derivative of the Right Side

First, take the derivative of the right side of the equation \(x^{4}+\frac{1}{x}+C\) using the power rule for derivatives. This gives the derivative of \(x^{4}\) as \(4x^{3}\) , the derivative of \(\frac{1}{x}\) as \(-\frac{1}{x^{2}}\), and the derivative of a constant (\(C\)) as zero, leading to: \(4x^{3}-\frac{1}{x^{2}}\)

Step 2: Compare the Resulting Derivative with the Integrand

Now, compare this derivative with the original integrand on the left side of the equation, which is \((4x^{3}-\frac{1}{x^{2}})\). In this case, the resulting derivative from Step 1 is identical to the integrand.

Step 3: Confirm the Statement

Therefore, because the derivative of the right side corresponds exactly to the integrand of the left side, the given initial statement is proved to be correct.