Solve ∫ (from 0 to 3) of (1/3x^2+2/3)

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

Evaluate

$$5$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 \frac{1}{3} times \frac{x^{3}}{3}.

$$\frac{x^{3}}{9}+\int \frac{2}{3}\mathrm{d}x$$

Find the integral of \frac{2}{3} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{9}+\frac{2x}{3}$$

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.

$$\frac{3^{3}}{9}+\frac{2}{3}\times 3-\left(\frac{0^{3}}{9}+\frac{2}{3}\times 0\right)$$

Simplify.

$$5$$