Evaluate the integral of x⁵(1 − x)⁴ from 0 to 1. Give the answer as a fraction in lowest terms.

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Evaluate the integral of $x^{5} (1 - x)^{4}$ from $0$ to $1$ . Give the answer as a fraction in lowest terms.

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Hint

Expanding the bracket works but is slow; look for a formula for integrals of this shape.

Solution

Use the Beta integral: the integral of $x^{m} (1 - x)^{n}$ from $0$ to $1$ equals $\frac{m ! n !}{( m + n + 1 )!}$ .
It follows by integrating by parts $n$ times, each time moving a power from $(1 - x)$ onto $x$ .
With $m = 5$ and $n = 4$ , this gives $\frac{5 ! \times 4 !}{10 !} = \frac{2880}{3628800}$ .
Reducing, the integral equals $\frac{2880}{3628800} = \frac{1}{1260}$ .

Answer: $\frac{1}{1260}$