Solve ∫ (1+4x-6x^2) wrt x

$$\int{ (1+4x-6 { x }^{ 2 } ) }d x$$

Evaluate

$$С+x+2x^{2}-2x^{3}$$

Solution Steps

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int 4x\mathrm{d}x+\int -6x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x+4\int x\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

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

$$x+4\int x\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

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}.

$$x+2x^{2}-6\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -6 times \frac{x^{3}}{3}.

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

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.

$$x+2x^{2}-2x^{3}+С$$

Differentiate w.r.t. x

$$1+4x-6x^{2}$$