Solve ∫ (from 1 to 4) of 5x-x^2-4

$$\int _ { 1 } ^ { 4 } 5 x - x ^ { 2 } - 4$$

Evaluate

$$\frac{9}{2}=4.5$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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}.

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

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

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

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.

$$\frac{5}{2}\times 4^{2}-\frac{4^{3}}{3}-4\times 4-\left(\frac{5}{2}\times 1^{2}-\frac{1^{3}}{3}-4\right)$$

Simplify.

$$\frac{9}{2}$$