Question

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

Short 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-by-step solution

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: \(4x^{3}-\frac{1}{x^{2}}\)

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.

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.