Solve =frac{sqrt{3}}{2}(5x-1/3x^3)|_1^2=frac{4sqrt{3}}{3}(J)

$$= \frac { \sqrt { 3 } } { 2 } ( 5 x - \frac { 1 } { 3 } x ^ { 3 } ) | _ { 1 } ^ { 2 } = \frac { 4 \sqrt { 3 } } { 3 } ( J )$$

Evaluate

$$\frac{\sqrt{3}x\left(15-x^{2}\right)}{6}$$

Short Solution Steps

Use the distributive property to multiply \frac{\sqrt{3}}{2} by 5x-\frac{1}{3}x^{3}.

$$5\times \frac{\sqrt{3}}{2}x-\frac{1}{3}\times \frac{\sqrt{3}}{2}x^{3}$$

Express 5\times \frac{\sqrt{3}}{2} as a single fraction.

$$\frac{5\sqrt{3}}{2}x-\frac{1}{3}\times \frac{\sqrt{3}}{2}x^{3}$$

Express \frac{5\sqrt{3}}{2}x as a single fraction.

$$\frac{5\sqrt{3}x}{2}-\frac{1}{3}\times \frac{\sqrt{3}}{2}x^{3}$$

Multiply -\frac{1}{3} times \frac{\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\sqrt{3}x}{2}+\frac{-\sqrt{3}}{3\times 2}x^{3}$$

Express \frac{-\sqrt{3}}{3\times 2}x^{3} as a single fraction.

$$\frac{5\sqrt{3}x}{2}+\frac{-\sqrt{3}x^{3}}{3\times 2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3\times 2 is 2\times 3. Multiply \frac{5\sqrt{3}x}{2} times \frac{3}{3}.

$$\frac{3\times 5\sqrt{3}x}{2\times 3}+\frac{-\sqrt{3}x^{3}}{2\times 3}$$

Since \frac{3\times 5\sqrt{3}x}{2\times 3} and \frac{-\sqrt{3}x^{3}}{2\times 3} have the same denominator, add them by adding their numerators.

$$\frac{3\times 5\sqrt{3}x-\sqrt{3}x^{3}}{2\times 3}$$

Do the multiplications in 3\times 5\sqrt{3}x-\sqrt{3}x^{3}.

$$\frac{15\sqrt{3}x-\sqrt{3}x^{3}}{2\times 3}$$

Factor the expressions that are not already factored in \frac{15\sqrt{3}x-\sqrt{3}x^{3}}{2\times 3}.

$$\frac{\sqrt{3}x\left(-x^{2}+15\right)}{2\times 3}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{x\left(-x^{2}+15\right)}{2\sqrt{3}}$$

Rationalize the denominator of \frac{x\left(-x^{2}+15\right)}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{x\left(-x^{2}+15\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{x\left(-x^{2}+15\right)\sqrt{3}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{x\left(-x^{2}+15\right)\sqrt{3}}{6}$$

Use the distributive property to multiply x by -x^{2}+15.

$$\frac{\left(-x^{3}+15x\right)\sqrt{3}}{6}$$

Use the distributive property to multiply -x^{3}+15x by \sqrt{3}.

$$\frac{-x^{3}\sqrt{3}+15x\sqrt{3}}{6}$$

Factor

$$\frac{\sqrt{3}x\left(15-x^{2}\right)}{6}$$