Solve ∫ (from 2 to 5) of 5/sqrt{x wrt x

$$\int _ { 2 } ^ { 5 } \frac { 5 } { \sqrt { x } } d x$$

Evaluate

$$10\left(\sqrt{5}-\sqrt{2}\right)\approx 8.218544151$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{5}{\sqrt{x}}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$5\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

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.

$$10\sqrt{x}$$

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.

$$10\times 5^{\frac{1}{2}}-10\times 2^{\frac{1}{2}}$$

Simplify.

$$10\sqrt{5}-10\sqrt{2}$$