Solve ∫ _{1/2}^5x^2+6x

$$\int _ { \frac { 1 } { 2 } } ^ { 5 } x ^ { 2 } + 6 x$$

Evaluate

$$\frac{927}{8}=115.875$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+6x\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 6x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x+6\int x\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+6\int x\mathrm{d}x$$

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

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

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.

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

Simplify.

$$\frac{927}{8}$$