Solve ∫ (from 0.5 to 1) of p^7(1-p)dp

$$\int _ { 0.5 } ^ { 1 } p ^ { 7 } ( 1 - p ) d p$$

Evaluate

$$\frac{251}{18432}\approx 0.013617622$$

Solution Steps

Use the distributive property to multiply p^{7} by 1-p.

$$\int _{0.5}^{1}p^{7}-p^{8}\mathrm{d}p$$

Evaluate the indefinite integral first.

$$\int p^{7}-p^{8}\mathrm{d}p$$

Integrate the sum term by term.

$$\int p^{7}\mathrm{d}p+\int -p^{8}\mathrm{d}p$$

Factor out the constant in each of the terms.

$$\int p^{7}\mathrm{d}p-\int p^{8}\mathrm{d}p$$

Since \int p^{k}\mathrm{d}p=\frac{p^{k+1}}{k+1} for k\neq -1, replace \int p^{7}\mathrm{d}p with \frac{p^{8}}{8}.

$$\frac{p^{8}}{8}-\int p^{8}\mathrm{d}p$$

Since \int p^{k}\mathrm{d}p=\frac{p^{k+1}}{k+1} for k\neq -1, replace \int p^{8}\mathrm{d}p with \frac{p^{9}}{9}. Multiply -1 times \frac{p^{9}}{9}.

$$\frac{p^{8}}{8}-\frac{p^{9}}{9}$$

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.

$$\frac{1^{8}}{8}-\frac{1^{9}}{9}-\left(\frac{0.5^{8}}{8}-\frac{0.5^{9}}{9}\right)$$

Simplify.

$$\frac{251}{18432}$$