Solve ∫ 4x*(x^2-4)-(2x^2-3)*2x

$$\int 4 x \times ( x ^ { 2 } - 4 ) - ( 2 x ^ { 2 } - 3 ) \times 2 x$$

Evaluate

$$С-5x^{2}$$

Solution Steps

Use the distributive property to multiply 4x by x^{2}-4.

$$\int 4x^{3}-16x-\left(2x^{2}-3\right)\times 2x\mathrm{d}x$$

Use the distributive property to multiply 2x^{2}-3 by 2.

$$\int 4x^{3}-16x-\left(4x^{2}-6\right)x\mathrm{d}x$$

Use the distributive property to multiply 4x^{2}-6 by x.

$$\int 4x^{3}-16x-\left(4x^{3}-6x\right)\mathrm{d}x$$

To find the opposite of 4x^{3}-6x, find the opposite of each term.

$$\int 4x^{3}-16x-4x^{3}+6x\mathrm{d}x$$

Combine 4x^{3} and -4x^{3} to get 0.

$$\int -16x+6x\mathrm{d}x$$

Combine -16x and 6x to get -10x.

$$\int -10x\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$-10\int x\mathrm{d}x$$

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 -10 times \frac{x^{2}}{2}.

$$-5x^{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-5x^{2}+С$$

Differentiate w.r.t. x

$$-10x$$