Solve ∫ (from 1 to 9) of sqrt{11/x wrt x

$$\int _ { 1 } ^ { 9 } \sqrt { \frac { 11 } { x } } d x$$

Evaluate

$$4\sqrt{11}\approx 13.266499161$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{\frac{11}{x}}\mathrm{d}x$$

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

$$\sqrt{11}\int \sqrt{\frac{1}{x}}\mathrm{d}x$$

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.

$$\sqrt{11}\times 2\sqrt{x}$$

Simplify.

$$2\sqrt{11}\sqrt{x}$$

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.

$$2\times 11^{\frac{1}{2}}\times 9^{\frac{1}{2}}-2\times 11^{\frac{1}{2}}\times 1^{\frac{1}{2}}$$

Simplify.

$$4\sqrt{11}$$