First slide

Evaluation of definite integrals

Question

$\int_{0}^{1} x {(1 - x)}^{n} d x =$

Moderate

A

$\frac{1}{n + 1}$
B

$\frac{1}{n + 2}$
C

$\frac{1}{(n + 1) (n + 2)}$
D

$\frac{2 n + 3}{(n + 1) (n + 2)}$

Solution

$\int_{0}^{1} x {(1 - x)}^{n} d x, x \to 1 - x$
$= \int_{0}^{1} (1 - x) x^{n} d x = \int_{0}^{1} (x^{n} - x^{n + 1}) d x$

$= \frac{1}{n + 1} - \frac{1}{n + 2} = \frac{1}{(n + 1) (n + 2)}$

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