Solve [ ( x ^ 2 - 1 ) 2 x - ∫ - (1/2x-3)]x

$$[ ( x ^ { 2 } - 1 ) 2 x - \int - ( \frac { 1 } { 2 } x - 3 ) ] x$$

Differentiate w.r.t. x

$$8x^{3}+\frac{3x^{2}}{4}-С-10x$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply x^{2}-1 by 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\left(2x^{2}-2\right)x-\int -\left(\frac{1}{2}x-3\right)\mathrm{d}x\right)x)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(2x^{3}-2x-\int -\left(\frac{1}{2}x-3\right)\mathrm{d}x\right)x)$$

To find the opposite of \frac{1}{2}x-3, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(2x^{3}-2x-\int -\frac{1}{2}x+3\mathrm{d}x\right)x)$$

Use the distributive property to multiply 2x^{3}-2x-\int -\frac{1}{2}x+3\mathrm{d}x by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{4}-2x^{2}-\int -\frac{1}{2}x+3\mathrm{d}xx)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$4\times 2x^{4-1}+2\left(-2\right)x^{2-1}+\left(-\int -\frac{x}{2}+3\mathrm{d}x\right)x^{1-1}$$

Multiply 4 times 2.

$$8x^{4-1}+2\left(-2\right)x^{2-1}+\left(-\int -\frac{x}{2}+3\mathrm{d}x\right)x^{1-1}$$

Subtract 1 from 4.

$$8x^{3}+2\left(-2\right)x^{2-1}+\left(-\int -\frac{x}{2}+3\mathrm{d}x\right)x^{1-1}$$

Multiply 2 times -2.

$$8x^{3}-4x^{2-1}+\left(-\int -\frac{x}{2}+3\mathrm{d}x\right)x^{1-1}$$

Subtract 1 from 2.

$$8x^{3}-4x^{1}+\left(-\int -\frac{x}{2}+3\mathrm{d}x\right)x^{1-1}$$

Subtract 1 from 1.

$$8x^{3}-4x^{1}+\left(-\left(-\frac{x^{2}}{4}+3x+С\right)\right)x^{0}$$

For any term t, t^{1}=t.

$$8x^{3}-4x+\left(-\left(-\frac{x^{2}}{4}+3x+С\right)\right)x^{0}$$

For any term t except 0, t^{0}=1.

$$8x^{3}-4x+\left(-\left(-\frac{x^{2}}{4}+3x+С\right)\right)\times 1$$

For any term t, t\times 1=t and 1t=t.

$$8x^{3}-4x-\left(-\frac{x^{2}}{4}+3x+С\right)$$

Evaluate

$$\frac{x\left(8x^{3}+x^{2}-20x\right)}{4}-Сx$$