Solve ∫ x^2(1-x) wrt x

$$\int x ^ { 2 } ( 1 - x ) d x$$

Evaluate

$$-\frac{x^{4}}{4}+\frac{x^{3}}{3}+С$$

Solution Steps

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

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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 -1 times \frac{x^{4}}{4}.

$$\frac{x^{3}}{3}-\frac{x^{4}}{4}$$

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.

$$\frac{x^{3}}{3}-\frac{x^{4}}{4}+С$$

Differentiate w.r.t. x

$$\left(1-x\right)x^{2}$$