Solve ∫ 5x(2x^2-4) wrt x

$$\int{ 5x(2 { x }^{ 2 } -4) }d x$$

Evaluate

$$\frac{5x^{4}}{2}-10x^{2}+С$$

Solution Steps

Use the distributive property to multiply 5x by 2x^{2}-4.

$$\int 10x^{3}-20x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 10x^{3}\mathrm{d}x+\int -20x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$10\int x^{3}\mathrm{d}x-20\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 10 times \frac{x^{4}}{4}.

$$\frac{5x^{4}}{2}-20\int x\mathrm{d}x$$

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 -20 times \frac{x^{2}}{2}.

$$\frac{5x^{4}}{2}-10x^{2}$$

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.

$$\frac{5x^{4}}{2}-10x^{2}+С$$

Differentiate w.r.t. x

$$10x\left(x^{2}-2\right)$$