Solve ∫ _{-3/2}^1(-1/2x+3/2) wrt x

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

Evaluate

$$\frac{65}{16}=4.0625$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$-\frac{x^{2}}{4}+\int \frac{3}{2}\mathrm{d}x$$

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

$$-\frac{x^{2}}{4}+\frac{3x}{2}$$

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.

$$-\frac{1^{2}}{4}+\frac{3}{2}\times 1-\left(-\frac{\left(-\frac{3}{2}\right)^{2}}{4}+\frac{3}{2}\left(-\frac{3}{2}\right)\right)$$

Simplify.

$$\frac{65}{16}$$