First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{1} x^{2} (1 - x)^{9} d x$ is

Easy

A

$\frac{1}{610}$
B

$\frac{1}{630}$
C

$\frac{1}{640}$
D

$\frac{1}{660}$

Solution

$\begin{array}{l} \int_{0}^{1} x^{2} (1 - x)^{9} d x = - \int_{1}^{0} (1 - t)^{2} t^{9} d t (1 - x = t) \\ = \int_{0}^{1} (1 - t^{2} - 2 t) t^{9} d t \\ = \frac{t^{10}}{10} + \frac{t^{12}}{12} - {\frac{2 t^{11}}{11}|}_{0}^{1} \\ = \frac{1}{10} + \frac{1}{12} - \frac{2}{11} = \frac{1}{660} \end{array}$

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