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

$$\int _ { 1 } ^ { 3 } x ^ { 2 } + x + 1 d x$$

Evaluate

$$\frac{44}{3}\approx 14.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+x+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\frac{x^{2}}{2}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

$$\frac{44}{3}$$