Solve ∫ (from 2 to 3) of (sqrt{x}+1/sqrt{x})

$$\int _ { 2 } ^ { 3 } ( \sqrt { x } + \frac { 1 } { \sqrt { x } } )$$

Evaluate

$$-\frac{10\sqrt{2}}{3}+4\sqrt{3}\approx 2.214158022$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Rewrite \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{3}{2}}}{\frac{3}{2}}. Simplify.

$$\frac{2x^{\frac{3}{2}}}{3}+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

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.

$$\frac{2x^{\frac{3}{2}}}{3}+2\sqrt{x}$$

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.

$$\frac{2}{3}\times 3^{\frac{3}{2}}+2\times 3^{\frac{1}{2}}-\left(\frac{2}{3}\times 2^{\frac{3}{2}}+2\times 2^{\frac{1}{2}}\right)$$

Simplify.

$$4\sqrt{3}-\frac{10\sqrt{2}}{3}$$