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Chapter 5: Problem 6

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

Short Answer

Expert verified
The original statement is verified, as the derivative of the function on the right side, \(x^2 - 4\), is identical to the simplified form of the integrand on the left side, \(x^2 - 4\).

Step by step solution

01

Identify the function to differentiate

The function to be differentiated is on the right side of the equation. This is \(\frac{1}{3}x^3 -4x + C\).
02

Differentiate the function

Apply the power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\), and the derivative of a constant is zero. Thus the derivative of the function is \(x^2 - 4\).
03

Simplify the integrand for comparison

The integrand on the left side of the equation is \((x-2)(x+2)\). Multiplication of the terms gives \(x^2 - 4\).
04

Compare the derivative and the integrand

We now have the derivative from the right side, \(x^2-4\), and the simplified form of the integrand from the left side, \(x^2 -4\). As these are the same, the original statement is verified.

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