Question

What is wrong with the equation?

\(\) \(\int\limits_{ - 1}^3 {\frac{1}{{{x^2}}}dx = \left( {\frac{{{x^{ - 1}}}}{{ - 1}}} \right)_{ - 1}^3 = - \frac{4}{3}} \)

Short answer

We cannot apply the evaluation theorem

Step-by-step solution

Step 1: What’s given in the Question.

Given that the function is\(( - 1,3)\)\(f(x) = \frac{1}{{{x^2}}}\)

Here \(\int\limits_{{\rm{ - 1}}}^{\rm{3}} {\frac{{\rm{1}}}{{{{\rm{x}}^{\rm{2}}}}}} {\rm{dx}}\) does not exist because the function \(f(x) = \frac{1}{{{x^2}}}\)has an infinite discontinuity at \({\rm{x = 3}}\) and \({\rm{x = - 1}}\)

Checking for Discontinuity.

That is,\({\rm{f}}\)is discontinuous on the interval\(( - 1,3)\)

Since \({\rm{f}}\)is discontinuous, we cannot apply the evaluation theorem.