What is the integral from 0 to 1 of x^5(1-x)^3 ?

The integral from 0 to 1 of x^5(1-x)^3 is 1/504

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integral from 0 to 1 of x^5(1-x)^3

\int_{0}^{1} x^{5} (1 - x)^{3} d x

\frac{1}{504}

Decimal

0.00198 \dots

Solution steps

\int_{0}^{1} x^{5} (1 - x)^{3} d x

Expand

x^{5} (1 - x)^{3} : x^{5} - 3 x^{6} + 3 x^{7} - x^{8}

= \int_{0}^{1} x^{5} - 3 x^{6} + 3 x^{7} - x^{8} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} x^{5} d x - \int_{0}^{1} 3 x^{6} d x + \int_{0}^{1} 3 x^{7} d x - \int_{0}^{1} x^{8} d x

\int_{0}^{1} x^{5} d x = \frac{1}{6}

\int_{0}^{1} 3 x^{6} d x = \frac{3}{7}

\int_{0}^{1} 3 x^{7} d x = \frac{3}{8}

\int_{0}^{1} x^{8} d x = \frac{1}{9}

= \frac{1}{6} - \frac{3}{7} + \frac{3}{8} - \frac{1}{9}

Simplify

\frac{1}{6} - \frac{3}{7} + \frac{3}{8} - \frac{1}{9} : \frac{1}{504}

= \frac{1}{504}

What is the integral from 0 to 1 of x^5(1-x)^3 ?
The integral from 0 to 1 of x^5(1-x)^3 is 1/504