Solve ∫ (from 4 to 9) of 1/2(x^2-2) wrt x

$$\int _ { 4 } ^ { 9 } \frac { 1 } { 2 } ( x ^ { 2 } - 2 ) d x$$

Evaluate

$$\frac{635}{6}\approx 105.833333333$$

Solution Steps

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

$$\int _{4}^{9}\frac{1}{2}x^{2}-1\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

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

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.

$$\frac{9^{3}}{6}-9-\left(\frac{4^{3}}{6}-4\right)$$

Simplify.

$$\frac{635}{6}$$