Integral of One Over Square Root of X from Zero to Nine

Evaluate the integral $\displaystyle \int_{0}^{9} \frac{1}{\sqrt{x}} \, dx$.

Evaluating the integral of one over the square root of x from zero to nine requires a firm understanding of definite integrals and improper integrals. This integral poses a characteristic challenge where the integrand becomes unbounded at the lower limit of the integral, x equals zero, making it an improper integral. When faced with such issues, the approach involves considering the limit where x approaches zero from the positive side. This ensures that the integration process appropriately handles the discontinuity at the boundary.

The antiderivative involved usually includes handling of fractional exponents and requires a keen eye to the properties of power functions. In this scenario, since the power of x is negative, specific methods of integration are used to transform and simplify the integrand into a more workable form. The simplicity of the integrand, 1 over the square root of x, allows for the usage of fundamental integration techniques, where one draws on the basic formula of integrating x raised to a power.

It's crucial to remember that evaluating definite integrals, especially those that appear improper, is central to understanding the broader applications of calculus, including computing areas, volumes, and more. By mastering this type of integral, you lay the groundwork for evaluating more complex integrals and contribute meaningfully to solving real-world problems with calculus. This problem is an excellent starter to develop skills in evaluating integrals where the function approaches infinity at one or more endpoints of the integration interval.

Related Problems