Solve ∫ x^2-x^3/x wrt x

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

Evaluate

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

Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-x^{3}}{x}.

$$\int \frac{\left(-x+1\right)x^{2}}{x}\mathrm{d}x$$

Cancel out x in both numerator and denominator.

$$\int x\left(-x+1\right)\mathrm{d}x$$

Expand the expression.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

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

Simplify.

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

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.

$$\frac{x^{2}}{2}-\frac{x^{3}}{3}+С$$

Differentiate w.r.t. x

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