Question - Evaluation of an Improper Integral

The image shows an improper integral of the form \(\int \sqrt{x} dx\). To solve this integral, we use the power rule for integration. The power rule states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), for any real number \(n\) not equal to -1, where \(C\) is the constant of integration.

The integral can thus be solved as follows:

\begin{align*} \int \sqrt{x} dx &= \int x^{1/2} dx \\ &= \frac{x^{(1/2) + 1}}{(1/2) + 1} + C \\ &= \frac{x^{3/2}}{3/2} + C \\ &= \frac{2}{3} x^{3/2} + C. \end{align*}

Therefore, the solution to the integral is \(\frac{2}{3} x^{3/2} + C\).