Solve ∫ (from 1 to 2) of x-sqrt{1/x wrt x

$$\int_{ 1 }^{ 2 } x- \sqrt{ \frac{ 1 }{ x } } d x$$

Evaluate

$$\frac{7}{2}-2\sqrt{2}\approx 0.671572875$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

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

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. Multiply -1 times 2\sqrt{x}.

$$\frac{x^{2}}{2}-2\sqrt{x}$$

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.

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

Simplify.

$$\frac{7}{2}-2\sqrt{2}$$