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

Step 1 ：Let's evaluate the improper integral \(\int_{1}^{\infty} x^{-\frac{1}{2}} dx\).

Step 2 ：The general formula for the integral of a power function is \(\int x^n dx = \frac{1}{n+1}x^{n+1} + C\), where \(n \neq -1\).

Step 3 ：In this case, the upper limit of the integral is infinity, which makes it an improper integral. We can solve this by taking the limit as the upper limit approaches infinity.

Step 4 ：Let's apply the formula to our integral. We get \(F = 2.0*x^{0.5}\).

Step 5 ：Next, we calculate the limit of \(F\) as \(x\) approaches infinity. The result of the limit calculation is infinity.

Step 6 ：This means the integral diverges. Therefore, the integral \(\int_{1}^{\infty} x^{-\frac{1}{2}} dx\) does not have a finite value.

Step 7 ：\(\boxed{\text{The integral }\int_{1}^{\infty} x^{-\frac{1}{2}} dx\text{ diverges.}}\)