Solve ∫ x^3-4x^2+3x/x-x^2 wrt x

$$\int \frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x } { x - x ^ { 2 } } d x$$

Evaluate

$$-\frac{x^{2}}{2}+3x+С$$

Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}-4x^{2}+3x}{x-x^{2}}.

$$\int \frac{x\left(x-3\right)\left(x-1\right)}{x\left(-x+1\right)}\mathrm{d}x$$

Extract the negative sign in -1+x.

$$\int \frac{-x\left(x-3\right)\left(-x+1\right)}{x\left(-x+1\right)}\mathrm{d}x$$

Cancel out x\left(-x+1\right) in both numerator and denominator.

$$\int -\left(x-3\right)\mathrm{d}x$$

Expand the expression.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 -1 times \frac{x^{2}}{2}.

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

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

Simplify.

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

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.

$$3x-\frac{x^{2}}{2}+С$$

Differentiate w.r.t. x

$$3-x$$