Solve ∫ 2x-x^2-x^2 wrt x

$$\int 2 x - x ^ { 2 } - x ^ { 2 } d x$$

Evaluate

$$-\frac{2x^{3}}{3}+x^{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

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

Simplify.

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

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.

$$x^{2}-\frac{2x^{3}}{3}+С$$

Differentiate w.r.t. x

$$2x\left(1-x\right)$$