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

$$\int _ { 1 } ^ { 3 } \frac { 1 } { 2 } ( x ^ { 2 } - 1 ) d x$$

Evaluate

$$\frac{10}{3}\approx 3.333333333$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by x^{2}-1.

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

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 \frac{1}{2} times \frac{x^{3}}{3}.

$$\frac{x^{3}}{6}+\int -\frac{1}{2}\mathrm{d}x$$

Find the integral of -\frac{1}{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{6}-\frac{x}{2}$$

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.

$$\frac{3^{3}}{6}-\frac{1}{2}\times 3-\left(\frac{1^{3}}{6}-\frac{1}{2}\right)$$

Simplify.

$$\frac{10}{3}$$