Solve ∫ (from 2 to 3) of 6x^2-2x+7 wrt x

$$\int_{ 2 }^{ 3 } 6 { x }^{ 2 } -2x+7 d x$$

Evaluate

$$40$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6x^{2}-2x+7\mathrm{d}x$$

Integrate the sum term by term.

$$\int 6x^{2}\mathrm{d}x+\int -2x\mathrm{d}x+\int 7\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x+\int 7\mathrm{d}x$$

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}.

$$2x^{3}-2\int x\mathrm{d}x+\int 7\mathrm{d}x$$

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}.

$$2x^{3}-x^{2}+\int 7\mathrm{d}x$$

Find the integral of 7 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x^{3}-x^{2}+7x$$

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.

$$2\times 3^{3}-3^{2}+7\times 3-\left(2\times 2^{3}-2^{2}+7\times 2\right)$$

Simplify.

$$40$$