Solve ∫ (from 4 to 9) of 1/sqrt{x wrt x

$$\int _ { 4 } ^ { 9 } \frac { 1 } { \sqrt { x } } d x$$

Evaluate

$$2$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Rewrite \frac{1}{\sqrt{x}} as x^{-\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form.

$$2\sqrt{x}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$2\times 9^{\frac{1}{2}}-2\times 4^{\frac{1}{2}}$$

Simplify.

$$2$$