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

$$\int _ { 1 } ^ { 2 } ( \frac { 1 } { \sqrt { x } } - x ) d x$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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.

$$2\sqrt{x}-\int x\mathrm{d}x$$

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}. Multiply -1 times \frac{x^{2}}{2}.

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

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.

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

Simplify.

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