Solve ∫ (from 5 to 9) of x^4-28x^3+286x^2-1260x+2025 wrt x

$$\int_{ 5 }^{ 9 } { x }^{ 4 } -28 { x }^{ 3 } +286 { x }^{ 2 } -1260x+2025 d x$$

Evaluate

$$\frac{512}{15}\approx 34.133333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{4}-28x^{3}+286x^{2}-1260x+2025\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{4}\mathrm{d}x+\int -28x^{3}\mathrm{d}x+\int 286x^{2}\mathrm{d}x+\int -1260x\mathrm{d}x+\int 2025\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{4}\mathrm{d}x-28\int x^{3}\mathrm{d}x+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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}.

$$\frac{x^{5}}{5}-28\int x^{3}\mathrm{d}x+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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 -28 times \frac{x^{4}}{4}.

$$\frac{x^{5}}{5}-7x^{4}+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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 286 times \frac{x^{3}}{3}.

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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 -1260 times \frac{x^{2}}{2}.

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-630x^{2}+\int 2025\mathrm{d}x$$

Find the integral of 2025 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-630x^{2}+2025x$$

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.

$$\frac{9^{5}}{5}-7\times 9^{4}+\frac{286}{3}\times 9^{3}-630\times 9^{2}+2025\times 9-\left(\frac{5^{5}}{5}-7\times 5^{4}+\frac{286}{3}\times 5^{3}-630\times 5^{2}+2025\times 5\right)$$

Simplify.

$$\frac{512}{15}$$