Solve ∫ (from 1 to 5) of (-x^4+8x^3-22x^2+40x-25) wrt x

$$\int _ { 1 } ^ { 5 } ( - x ^ { 4 } + 8 x ^ { 3 } - 22 x ^ { 2 } + 40 x - 25 ) d x$$

Evaluate

$$\frac{1408}{15}\approx 93.866666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -x^{4}+8x^{3}-22x^{2}+40x-25\mathrm{d}x$$

Integrate the sum term by term.

$$\int -x^{4}\mathrm{d}x+\int 8x^{3}\mathrm{d}x+\int -22x^{2}\mathrm{d}x+\int 40x\mathrm{d}x+\int -25\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\int x^{4}\mathrm{d}x+8\int x^{3}\mathrm{d}x-22\int x^{2}\mathrm{d}x+40\int x\mathrm{d}x+\int -25\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -1 times \frac{x^{5}}{5}.

$$-\frac{x^{5}}{5}+8\int x^{3}\mathrm{d}x-22\int x^{2}\mathrm{d}x+40\int x\mathrm{d}x+\int -25\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 8 times \frac{x^{4}}{4}.

$$-\frac{x^{5}}{5}+2x^{4}-22\int x^{2}\mathrm{d}x+40\int x\mathrm{d}x+\int -25\mathrm{d}x$$

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 -22 times \frac{x^{3}}{3}.

$$-\frac{x^{5}}{5}+2x^{4}-\frac{22x^{3}}{3}+40\int x\mathrm{d}x+\int -25\mathrm{d}x$$

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

$$-\frac{x^{5}}{5}+2x^{4}-\frac{22x^{3}}{3}+20x^{2}+\int -25\mathrm{d}x$$

Find the integral of -25 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{x^{5}}{5}+2x^{4}-\frac{22x^{3}}{3}+20x^{2}-25x$$

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.

$$-\frac{5^{5}}{5}+2\times 5^{4}-\frac{22}{3}\times 5^{3}+20\times 5^{2}-25\times 5-\left(-\frac{1^{5}}{5}+2\times 1^{4}-\frac{22}{3}\times 1^{3}+20\times 1^{2}-25\right)$$

Simplify.

$$\frac{1408}{15}$$