Solve ∫ (from 0 to 3) of x^2-2x+10

$$\int _ { 0 } ^ { 3 } x ^ { 2 } - 2 x + 10$$

Evaluate

$$30$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-2x+10\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -2x\mathrm{d}x+\int 10\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x+\int 10\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-2\int x\mathrm{d}x+\int 10\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-x^{2}+\int 10\mathrm{d}x$$

Find the integral of 10 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}-x^{2}+10x$$

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.

$$\frac{3^{3}}{3}-3^{2}+10\times 3-\left(\frac{0^{3}}{3}-0^{2}+10\times 0\right)$$

Simplify.

$$30$$