Solve ∫ _{3/2}^frac{9{2}}5x-x^2-9/4 wrt x

$$\int _ { \frac { 3 } { 2 } } ^ { \frac { 9 } { 2 } } 5 x - x ^ { 2 } - \frac { 9 } { 4 } d x$$

Evaluate

$$9$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5x-x^{2}-\frac{9}{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x\mathrm{d}x+\int -x^{2}\mathrm{d}x+\int -\frac{9}{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x\mathrm{d}x-\int x^{2}\mathrm{d}x+\int -\frac{9}{4}\mathrm{d}x$$

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 5 times \frac{x^{2}}{2}.

$$\frac{5x^{2}}{2}-\int x^{2}\mathrm{d}x+\int -\frac{9}{4}\mathrm{d}x$$

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

$$\frac{5x^{2}}{2}-\frac{x^{3}}{3}+\int -\frac{9}{4}\mathrm{d}x$$

Find the integral of -\frac{9}{4} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{5x^{2}}{2}-\frac{x^{3}}{3}-\frac{9x}{4}$$

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.

$$\frac{5}{2}\times \left(\frac{9}{2}\right)^{2}-\frac{\left(\frac{9}{2}\right)^{3}}{3}-\frac{9}{4}\times \frac{9}{2}-\left(\frac{5}{2}\times \left(\frac{3}{2}\right)^{2}-\frac{\left(\frac{3}{2}\right)^{3}}{3}-\frac{9}{4}\times \frac{3}{2}\right)$$

Simplify.

$$9$$