Solve ∫ x^2-1/x+1 wrt x

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

Evaluate

$$\frac{x^{2}}{2}-x+С$$

Solution Steps

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

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

Cancel out x+1 in both numerator and denominator.

$$\int x-1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int -1\mathrm{d}x$$

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}.

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

Find the integral of -1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$\frac{x^{2}}{2}-x+С$$

Differentiate w.r.t. x

$$x-1$$