Solve ∫ (-2x^3+2x) wrt x

$$\int ( - 2 x ^ { 3 } + 2 x ) d x$$

Evaluate

$$-\frac{x^{4}}{2}+x^{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 -2 times \frac{x^{4}}{4}.

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

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

$$-\frac{x^{4}}{2}+x^{2}$$

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.

$$-\frac{x^{4}}{2}+x^{2}+С$$

Differentiate w.r.t. x

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