Solve ∫ (from 2 to 6) of (18x^2-2x+2) wrt x

$$\int _ { 2 } ^ { 6 } ( 18 x ^ { 2 } - 2 x + 2 ) d x$$

Evaluate

$$1224$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 18x^{2}-2x+2\mathrm{d}x$$

Integrate the sum term by term.

$$\int 18x^{2}\mathrm{d}x+\int -2x\mathrm{d}x+\int 2\mathrm{d}x$$

Factor out the constant in each of the terms.

$$18\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

$$6x^{3}-2\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

$$6x^{3}-x^{2}+\int 2\mathrm{d}x$$

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

$$6x^{3}-x^{2}+2x$$

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.

$$6\times 6^{3}-6^{2}+2\times 6-\left(6\times 2^{3}-2^{2}+2\times 2\right)$$

Simplify.

$$1224$$