Question

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

Short answer

The derivative of the right-hand side of the equation is \(x^2 - 4\), which is equal to the integrand of the left-hand side, verifying the given integral equation.

Step-by-step solution

Recognize the Problem Type

The exercise is asking to verify an integral by using the Fundamental Theorem of Calculus, which basically says that the integral of a function f(x) from a to x is the anti-derivative F(x) of f. Therefore, to verify the given integral, the derivative of the right side of the equation, which is F(x), needs to be found and compared to f(x), which is (x-2)(x+2).

Find the Derivative of the Right Side

The right-hand side of the equation is the function F(x) = (1/3)*x^3 - 4x + C. The derivative of this function, denoted as F'(x), can be found using the power rule for derivatives, which states that the derivative of x^n is n*x^(n-1), and the constant rule, which says the derivative of a constant is zero. Therefore, F'(x) = (1/3)*3*x^(3-1) - 4 + 0 = x^2 - 4.

Compare F'(x) with f(x)

Now, the derivative of the right-hand side, F'(x) = x^2 - 4, is compared to the integrand of the left-hand side, f(x) = (x-2)(x+2). Notice that the expression (x-2)(x+2) is a difference of squares and can be simplified to x^2 - 2^2 = x^2 - 4. Therefore, F'(x) is indeed equal to f(x).