Solve ∫ (from 1 to 8) of x^2+5x

$$\int _ { 1 } ^ { 8 } x ^ { 2 } + 5 x$$

Evaluate

$$\frac{1967}{6}\approx 327.833333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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}.

$$\frac{x^{3}}{3}+\frac{5x^{2}}{2}$$

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.

$$\frac{8^{3}}{3}+\frac{5}{2}\times 8^{2}-\left(\frac{1^{3}}{3}+\frac{5}{2}\times 1^{2}\right)$$

Simplify.

$$\frac{1967}{6}$$