Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

18. The most general antiderivative of\(f\left( x \right) = {x^{ - 2}}\)is\(F\left( x \right) = \frac{{ - 1}}{x} + C\).

Short answer

The given statement is true.

Step-by-step solution

Antiderivative

The antiderivative of a function f is a function F such that \(\int {f\left( x \right)} dx = F\left( x \right) + C\).

Step 2: \(f'\) is periodic

Consider the given function as \(f\left( x \right) = {x^{ - 2}}\).

Find the antiderivative as follows:

\(\begin{array}{c}\int {f\left( x \right)} dx = \int {{x^{ - 2}}} dx\\ = \frac{{{x^{ - 2 + 1}}}}{{ - 2 + 1}} + c\\ = \frac{{{x^{ - 1}}}}{{ - 1}} + c\\ = \frac{{ - 1}}{x} + c\end{array}\)

Thus, the antiderivative is \(F\left( x \right) = \frac{{ - 1}}{x} + c\).

Therefore, the given statement is true.