Evaluate the improper integral $\int_{0}^{1} x^{-2} d x$

Step 1 ：We are asked to evaluate the improper integral \(\int_{0}^{1} x^{-2} d x\).

Step 2 ：This is an improper integral because the function \(x^{-2}\) is not defined at \(x=0\).

Step 3 ：We can rewrite the integral as \(\int_{0}^{1} \frac{1}{x^2} dx\) and then apply the power rule for integration, which states that \(\int x^n dx = \frac{1}{n+1}x^{n+1} + C\) for \(n \neq -1\). However, in this case, \(n = -2\), so we need to be careful when applying the rule.

Step 4 ：The result of the integral is infinity, which means the area under the curve of the function \(f(x) = x^{-2}\) from \(0\) to \(1\) is infinite. This makes sense because the function \(f(x) = x^{-2}\) approaches infinity as \(x\) approaches \(0\) from the right.

Step 5 ：Final Answer: The value of the improper integral \(\int_{0}^{1} x^{-2} d x\) is \(\boxed{\infty}\).