Solve ∫ (from 0 to 1) of (3/2sqrt{x}-2) wrt x

$$\int _ { 0 } ^ { 1 } ( \frac { 3 } { 2 } \sqrt { x } - 2 ) d x$$

Evaluate

$$-1$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{3\sqrt{x}}{2}-2\mathrm{d}x$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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. Multiply \frac{3}{2} times \frac{2x^{\frac{3}{2}}}{3}.

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

Find the integral of -2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{\frac{3}{2}}-2x$$

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.

$$1^{\frac{3}{2}}-2-\left(0^{\frac{3}{2}}-2\times 0\right)$$

Simplify.

$$-1$$