Solve ∫ x+2x^2/x wrt x

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

Evaluate

$$x^{2}+x+С$$

Solution Steps

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

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

Cancel out x in both numerator and denominator.

$$\int 2x+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int 1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}. Multiply 2 times \frac{x^{2}}{2}.

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

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

$$x^{2}+x$$

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.

$$x+x^{2}+С$$

Differentiate w.r.t. x

$$2x+1$$