Solve ∫ 1-2x+3

$$\int 1 - 2 x + 3$$

Evaluate

$$С+4x-x^{2}$$

Solution Steps

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int -2x\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x-2\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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

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

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

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

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

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

Simplify.

$$4x-x^{2}$$

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.

$$4x-x^{2}+С$$

Differentiate w.r.t. x

$$4-2x$$