Solve ∫ (from - 2 to 0) of (3x^2+x-1) wrt x

$$\int _ { - 2 } ^ { 0 } ( 3 x ^ { 2 } + x - 1 ) d x$$

Evaluate

$$4$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

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

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

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

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$0^{3}+\frac{0^{2}}{2}-0-\left(\left(-2\right)^{3}+\frac{\left(-2\right)^{2}}{2}-\left(-2\right)\right)$$

Simplify.

$$4$$