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

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

Evaluate

$$26$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Find the integral of -2 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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 3 times \frac{x^{3}}{3}.

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

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.

$$\frac{3^{2}}{2}-2\times 3+3^{3}-\left(\frac{1^{2}}{2}-2+1^{3}\right)$$

Simplify.

$$26$$