Solve ∫ x^2-5x^2+4x/x-1 wrt x

$$\int \frac { x ^ { 2 } - 5 x ^ { 2 } + 4 x } { x - 1 } d x$$

Evaluate

$$С-2x^{2}$$

Solution Steps

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$\int \frac{-4x^{2}+4x}{x-1}\mathrm{d}x$$

Factor the expressions that are not already factored in \frac{-4x^{2}+4x}{x-1}.

$$\int \frac{4x\left(-x+1\right)}{x-1}\mathrm{d}x$$

Extract the negative sign in 1-x.

$$\int \frac{-4x\left(x-1\right)}{x-1}\mathrm{d}x$$

Cancel out x-1 in both numerator and denominator.

$$\int -4x\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$-4\int x\mathrm{d}x$$

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

$$-2x^{2}$$

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.

$$-2x^{2}+С$$

Differentiate w.r.t. x

$$-4x$$